If you form the smash product $X = K(A,p) \wedge K(B,q)$ of two Eilenberg-MacLane spaces, then the resulting space is $(p+q-1)$-connected, and the first non-trivial homotopy group in dimension $p+q$ is $A \otimes B$.
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 $X$, 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 $(p+q+1)$-dimensional cells in $X$, you can compute $\pi_{p+q}$. Now you can use obstruction theory to produce a map $X \to K(A\otimes B, p+q)$.
