A certain submatrix of the correlation polytope
Every quadratic multilinear polynomial can be written as
$$
a_0 + \sum_i a_{ii} x_i + \sum_{i \neq j} a_{ij} x_i x_j =
a_0 + \sum_i a_{ii} x_i^2 + \sum_{i \neq j} a_{ij} x_i x_j,
$$
using the fact that $x_i^2 = x_i$ for every $x_i \in \{0,1\}$. This clearly equals $a_0 + \sum_{ij} A_{ij} x_i x_j$, which in turn is the same as what you wrote.

Regarding the matrix $M_n$, it is indeed infinite. For the proof that it is a submatrix of some slack matrix of the correlation polytope, see Lemma 2.8 of these lecture notes by James Lee.

If you have any more questions on the Bonn lecture notes, I suggest looking in the cited lecture notes, which seem to be more thorough.