I guess you have to read this in the context of intersection theory. On a manifold of dimension $2n$ we have the $n$-th homology group, which can informally be thought of as being generated by equivalence classes of $n$-dimensional submanifolds. Intersecting two of these should generally give a discrete set of points. Intuitively the intersection product can be thought of as the number of points in the intersection, but it is not exactly the same as the set-theoretic intersection.
To make this concrete, think of a genus $g$ Riemann surface, which has dimension 2. Its first cohomology group has rank $2g$, and as our generators $A$ we can take curves around the holes, and generators $B$ are curves cutting open a hole.
$A_i$ and $B_j$ intersect in a single point exactly when $i = j$, otherwise they don't. The anticommutative structure is part of the formalism that makes everything work out, and that is quite technical.