Why adding combinations of features would increase performance of linear SVM?
Multiplication is not a linear operation. Your linear SVM constructs a (hyper-)plane
$$
w_0 = w_1 x_1 + w_2 x_2
$$
for some weights $w_0, w_1, w_2.$

By introducing the AND-feature, you add another dimension:
$$
w_0 = w_1 x_1 + w_2 x_2 + w_3 x_1 x_2.
$$

It might well be that your two-dimensional data set is not linearly separable, but the three-dimensional data set is. 

A small addition: Would adding the OR-feature increase performance even further? No, because it is a linear combination of the other three features: $x \vee y = x + y - (x \wedge y)$ where $\vee$ is OR and $\wedge$ is AND.