If you form the smash product X = K(A,p) n K(B,q)$X = K(A,p) \wedge K(B,q)$ of two Eilenberg-MacLane spaces (I am using "n" for the smash product symbol here), then the resulting space is (p+q-1)$(p+q-1)$-connected, and the first non-trivial homotopy group in dimension p+q$p+q$ is A \otimes B$A \otimes B$.
To see this "geometrically", I would model the EM spaces as CW-complexes, whose first non-basepoint cells are in dimensions p and q respectively. Then in the smash product X$X$, the bottom dimensional cells are products of the bottom dimensional cells of the EM spaces; this gives the connectivity result, and by looking at the attaching maps of the (p+q+1)$(p+q+1)$-dimensional cells in X$X$, you can compute \pi_{p+q}$\pi_{p+q}$. Now you can use obstruction theory to produce a map X --> K(A\otimes B, p+q)$X \to K(A\otimes B, p+q)$.
