A short answer to the question the title asks would be: Because Clifford algebra in odd dimensions is not simple.
An extended and more understandable explanation obviously needs some calculations, that will relate Fierz identities in 11 dimensions to central charges of $\mathcal{N}=1$ superalgebra. And the reducibility argument will allow to include the correct amount of central charges.
Certainly, one should realise that the arguments below do not give an exhaustive answer to the question, one necessarily needs to consider solutions of EOM of 11 dimensional sugra, look at the realisation of superalgebra and susy multiplets. But these simple calculations certainly give a hint to at least intuitive understanding, that is obviously necessary for our work.
Start with a brief reminder of the Clifford construction. Consider a vector space $V$ with basis given by $d$ vectors $\{e_i\}$. denote the metric on this vector space by $g_{mn}$.
Clifford algebra $Cl(d)$ of the $d$-dimensional linear space $V$ is generated by all possible products of $e_i$. Basically, we now think of the basis vectors as elements of an algebra that can be multiplied. In addition the following anticommutation identity is imposed
\begin{equation}
\{e_i,e_j\}=2g_{ij}.
\end{equation}
If we choose the metric to be diagonal with eigenvalues $\pm 1$, then two elements $e_i$ and $e_j$ with $i\neq j$ anticommute. These are the well-known Dirac gamma matrices. In general, there are $2^{d}$ of them.
Basis of the Clifford algebra is then given by
\begin{equation}
bas(Cl(d))=\{1,e_i, e_ie_j, e_ie_je_k, \ldots\}.
\end{equation}
The subset of the generators given only by even product of gamma matrices itself forms an algebra that we denote by $Cl_0(d)$. This is the so-called even subalgebra of Clifford algebra.
Further discussion depends on whether the Clifford algebra or its even subalgebra is simple. To investigate this property we define the chirality operator that has the following form
\begin{equation}
\epsilon= e_1e_2\cdots e_d,
\end{equation}
i.e. it is just multiplication of all $e_i$'s. Obviously, its square $\epsilon^2$ is $\pm 1$ depending on the dimension and signature (the precise sign is not important for our discussion), that allows to define projector operators
\begin{equation}
P_{\pm}=\frac{1}{2}(1\pm \epsilon).
\end{equation}
We now ask: whether these projectors induce an invariant splitting of the Clifford algebra or its even subalgebra? It appears, that they do if the chirality operator is in the center of the corresponding algebra.
A center $Z$ of an algebra $A$ is defined as such a subset of elements of $A$, that commute with any element of $A$
\begin{equation}
Z=\{z\in A; [z,a]=0,\forall a\in A\}.
\end{equation}
A short calculation shows that a commutator of $\epsilon$ with an element $e_{k_1}\cdots e_{k_l}$ of the Clifford algebra $Cl(d)$ is proportional to
\begin{equation}
[\epsilon,e_{k_1}\cdots e_{k_l}] \propto (-1)^{l(d-1)}-1,
\end{equation}
the remained pieces on the LHS are not of our interest here.
Now we are to get the most important result of this post. For a even-dimensional vector space $d=2m$ the chirality operator is not in the center $Z$ of $Cl(2m)$ in general. But it is in the center $Z_0$ of its even subalgebra $Cl_0(2m)$, since then $l=2k$ is always even. This means, that the even subalgebra is not simple and can be written as
\begin{equation}
Cl_0(2m)=P_+Cl_0(2m)\oplus P_-Cl_0(2m).
\end{equation}
This is the famous decomposition of a Dirac spinor into 2 Weyl spinors. So, nothing amazing so far.
However, next one notes that for the odd-dimensional case the chirality operator is in the center $Z$ of $Cl(2d+1)$. Hence, the Clifford algebra itself can be decomposed
\begin{equation}
Cl(2m+1)=P_+Cl(2m+1)\oplus P_-Cl(2m+1).
\end{equation}
Moreover, one can show an amazing property $P_\pm Cl(2m+1)=Cl(2m)$. In other words, in odd dimensions the Clifford algebra is just doubled that of one dimension less.
This can be easely show on the simple example of $Cl(2)$ and $Cl(3)$. These have the following
\begin{equation}
\begin{aligned}
&bas Cl(2)=\{1, e_1, e_2, e_{12}\},\\
&bas Cl(3)=\{1, e_1, e_2, e_3, e_{12}, e_{13}, e_{23}, e_1e_2e_3\}.
\end{aligned}
\end{equation}
But, we know that $e_{12}$ is actually $e_3$ and $e_{13}=e_2$, $e_{23}=e_1$, in addition $e_1e_2e_3=1$. Hence, we see that $Cl(3)$ is the doubled version of $Cl(2)$. Of course, one should realise that these are different algebras since they have different representations and properties of spinors.
All in all, we can conclude that the matrices
\begin{equation}
e_{k_1\ldots k_l} = e_{[k_1}\cdots e_{k_l]}
\end{equation}
with $l=0\ldots d$, when $d$ is even, and $l=0\ldots \frac{d-1}{2}$, when $d$ is odd consistute a basis of the representation of $Cl(d)$.
Finally, we turn, to 11-dimensional supergravity and is supersymmetry algebra. Start with the familiar anticommutator $\{Q,\bar{Q}\}$, that is just a product of 2 Majorana spinors and thus can be decomposed according to Fierz identities. Following the usual Fierz procedure one should remember to count generators only up to $d-1/2=5$. Although, there are symmetric generators $e_{\mu_1\ldots \mu_l}$ with higher number of indices, taking them into account will just lead to double-counting.
We can use the result of this paper and write
\begin{equation}
\{Q,\bar{Q}\}= P^\mu \Gamma_\mu C_{11}^{-1} + \frac{1}{2!}Z^{\mu\nu}\Gamma_{\mu\nu}C_{11}^{-1} + \frac{1}{5!}Z^{\mu_1\mu_2\mu_3\mu_4\mu_5}\Gamma_{\mu_1\mu_2\mu_3\mu_4\mu_5}C_{11}^{-1},
\end{equation}
where $C_{11}$ is just the 11-dimensional charge conjugation operator. The central charges with 2 and 5 indices correspond to the M2 and M5 branes.
For further reading recommended are the very nice introductory paper by M. Rausch de Traubenberg , lectures by R. Coquereaux and of course the renowned book by Salamon "Spin geometry and Seiberg-Witten invariants". Finally, a pedagogical explanation of 11D SUGRA and its solitonic solutions can be found in the paper by Andre Miemiec and Igor Schnakenburg.
An extended and more understandable explanation obviously needs some calculations, that will relate Fierz identities in 11 dimensions to central charges of $\mathcal{N}=1$ superalgebra. And the reducibility argument will allow to include the correct amount of central charges.
Certainly, one should realise that the arguments below do not give an exhaustive answer to the question, one necessarily needs to consider solutions of EOM of 11 dimensional sugra, look at the realisation of superalgebra and susy multiplets. But these simple calculations certainly give a hint to at least intuitive understanding, that is obviously necessary for our work.
Start with a brief reminder of the Clifford construction. Consider a vector space $V$ with basis given by $d$ vectors $\{e_i\}$. denote the metric on this vector space by $g_{mn}$.
Clifford algebra $Cl(d)$ of the $d$-dimensional linear space $V$ is generated by all possible products of $e_i$. Basically, we now think of the basis vectors as elements of an algebra that can be multiplied. In addition the following anticommutation identity is imposed
\begin{equation}
\{e_i,e_j\}=2g_{ij}.
\end{equation}
If we choose the metric to be diagonal with eigenvalues $\pm 1$, then two elements $e_i$ and $e_j$ with $i\neq j$ anticommute. These are the well-known Dirac gamma matrices. In general, there are $2^{d}$ of them.
Basis of the Clifford algebra is then given by
\begin{equation}
bas(Cl(d))=\{1,e_i, e_ie_j, e_ie_je_k, \ldots\}.
\end{equation}
The subset of the generators given only by even product of gamma matrices itself forms an algebra that we denote by $Cl_0(d)$. This is the so-called even subalgebra of Clifford algebra.
Further discussion depends on whether the Clifford algebra or its even subalgebra is simple. To investigate this property we define the chirality operator that has the following form
\begin{equation}
\epsilon= e_1e_2\cdots e_d,
\end{equation}
i.e. it is just multiplication of all $e_i$'s. Obviously, its square $\epsilon^2$ is $\pm 1$ depending on the dimension and signature (the precise sign is not important for our discussion), that allows to define projector operators
\begin{equation}
P_{\pm}=\frac{1}{2}(1\pm \epsilon).
\end{equation}
We now ask: whether these projectors induce an invariant splitting of the Clifford algebra or its even subalgebra? It appears, that they do if the chirality operator is in the center of the corresponding algebra.
A center $Z$ of an algebra $A$ is defined as such a subset of elements of $A$, that commute with any element of $A$
\begin{equation}
Z=\{z\in A; [z,a]=0,\forall a\in A\}.
\end{equation}
A short calculation shows that a commutator of $\epsilon$ with an element $e_{k_1}\cdots e_{k_l}$ of the Clifford algebra $Cl(d)$ is proportional to
\begin{equation}
[\epsilon,e_{k_1}\cdots e_{k_l}] \propto (-1)^{l(d-1)}-1,
\end{equation}
the remained pieces on the LHS are not of our interest here.
Now we are to get the most important result of this post. For a even-dimensional vector space $d=2m$ the chirality operator is not in the center $Z$ of $Cl(2m)$ in general. But it is in the center $Z_0$ of its even subalgebra $Cl_0(2m)$, since then $l=2k$ is always even. This means, that the even subalgebra is not simple and can be written as
\begin{equation}
Cl_0(2m)=P_+Cl_0(2m)\oplus P_-Cl_0(2m).
\end{equation}
This is the famous decomposition of a Dirac spinor into 2 Weyl spinors. So, nothing amazing so far.
However, next one notes that for the odd-dimensional case the chirality operator is in the center $Z$ of $Cl(2d+1)$. Hence, the Clifford algebra itself can be decomposed
\begin{equation}
Cl(2m+1)=P_+Cl(2m+1)\oplus P_-Cl(2m+1).
\end{equation}
Moreover, one can show an amazing property $P_\pm Cl(2m+1)=Cl(2m)$. In other words, in odd dimensions the Clifford algebra is just doubled that of one dimension less.
This can be easely show on the simple example of $Cl(2)$ and $Cl(3)$. These have the following
\begin{equation}
\begin{aligned}
&bas Cl(2)=\{1, e_1, e_2, e_{12}\},\\
&bas Cl(3)=\{1, e_1, e_2, e_3, e_{12}, e_{13}, e_{23}, e_1e_2e_3\}.
\end{aligned}
\end{equation}
But, we know that $e_{12}$ is actually $e_3$ and $e_{13}=e_2$, $e_{23}=e_1$, in addition $e_1e_2e_3=1$. Hence, we see that $Cl(3)$ is the doubled version of $Cl(2)$. Of course, one should realise that these are different algebras since they have different representations and properties of spinors.
All in all, we can conclude that the matrices
\begin{equation}
e_{k_1\ldots k_l} = e_{[k_1}\cdots e_{k_l]}
\end{equation}
with $l=0\ldots d$, when $d$ is even, and $l=0\ldots \frac{d-1}{2}$, when $d$ is odd consistute a basis of the representation of $Cl(d)$.
Finally, we turn, to 11-dimensional supergravity and is supersymmetry algebra. Start with the familiar anticommutator $\{Q,\bar{Q}\}$, that is just a product of 2 Majorana spinors and thus can be decomposed according to Fierz identities. Following the usual Fierz procedure one should remember to count generators only up to $d-1/2=5$. Although, there are symmetric generators $e_{\mu_1\ldots \mu_l}$ with higher number of indices, taking them into account will just lead to double-counting.
We can use the result of this paper and write
\begin{equation}
\{Q,\bar{Q}\}= P^\mu \Gamma_\mu C_{11}^{-1} + \frac{1}{2!}Z^{\mu\nu}\Gamma_{\mu\nu}C_{11}^{-1} + \frac{1}{5!}Z^{\mu_1\mu_2\mu_3\mu_4\mu_5}\Gamma_{\mu_1\mu_2\mu_3\mu_4\mu_5}C_{11}^{-1},
\end{equation}
where $C_{11}$ is just the 11-dimensional charge conjugation operator. The central charges with 2 and 5 indices correspond to the M2 and M5 branes.
For further reading recommended are the very nice introductory paper by M. Rausch de Traubenberg , lectures by R. Coquereaux and of course the renowned book by Salamon "Spin geometry and Seiberg-Witten invariants". Finally, a pedagogical explanation of 11D SUGRA and its solitonic solutions can be found in the paper by Andre Miemiec and Igor Schnakenburg.
