Suppose $M=M_1 \oplus \dots \oplus M_r$ with $M_i$ indecomposable for $1 \leq i \leq r$. Let $\mathcal F (M_1, \dots, M_r)$ denote the category of modules which admit a filtration with subquotients in the $M_i$. The crucial assumption in Keller's proof is that $\mathcal F(M_1, \dots, M_r)$ is closed under syzygies.

With this assumption, any $n$-fold extension ($n>1$) $$\xi \in \operatorname{Ext}^n(M_u,M_v) \simeq \operatorname{Ext}^1(\Omega^{n-1}(M_u),M_v)$$ with $1 \leq u,v \leq r$ is the Yoneda splice of $n$ short-exact sequences $$0 \to X_{i+1} \to E_i \to X_i \to 0 \quad (0 \leq i \leq n-1)$$ where $X_0=M_u$, $X_n=M_v$, and each $X_i$ is an object in $\mathcal F(M_1, \dots, M_r)$ for $0 \leq i \leq n$.

As explained in the article you mentioned, the category $\mathcal F(M_1, \dots, M_r)$ is equivalent to a category of *twisted stalks*. A twisted stalk can be described as a pair $(B,\delta)$, where $B$ is a sequence of subfactors from $\{M_1, \dots, M_r\}$ and $\delta$ is an upper triangular matrix with entries from $\operatorname{Ext}^1(M,M)$.

If $X_i$ corresponds to a twisted stalk $(B,\delta)$ and $X_{i+1}$ corresponds to a twisted stalk $(B',\delta')$, then the short-exact sequence $0 \to X_{i+1} \to E_i \to X_i \to 0$ corresponds to a morphism of twisted *complexes* $(B,\delta) \to (B',\delta')[1]$. Such a morphism is a matrix with entries from $\operatorname{Ext}^1(M,M)$.

So each short-exact sequence $0 \to X_{i+1} \to E_i \to X_i \to 0$ can be described using only the $\operatorname{Ext}^1$ part of $\operatorname{Ext}^\ast(M,M)$. The higher multiplications enter the picture when we splice the short-exact sequences together. The rule for composing morphisms of twisted complexes involves $m_2$ and higher multiplications of the matrices involved. When splicing all the short-exact sequences together, we end up with an expression for $\xi \in \operatorname{Ext}^n(M_u,M_v)$ using only elements in $\operatorname{Ext}^1(M,M)$ and $m_2$ and higher multiplications. This shows that $\operatorname{Ext}^\ast(M,M)$ is generated in degree $0$ and $1$ as an $A_\infty$-algebra.

A class of examples where the assumption is satisfied is when $\{M_1, \dots, M_r\}$ are the standard modules over a quasi-hereditary algebra.