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,
But T(B)=B and T(A)=A because they are symmetric.
And hence they commute.