Let me proof it one way, that should show you how do the other direction.

I shall use "T" to represent the transpose.

By hypothesis A and B are symmetric.

Thus, T(A)=A and T(B)=B by definition.

We show that if AB is symmetric then AB commute.

That is T(AB)=AB

But by a theorem we know that,

T(AB)=T(B)T(A)

Thus,

T(B)T(A)=AB

But T(B)=B and T(A)=A because they are symmetric.

Thus,

BA=AB

And hence they commute.