Prove that A* is the smallest reflexive and transitive set containing A
Hint: Try the usual approach to show set inclusion! That is, pick $w \in A^*$ and show that $w \in B$.

Next hint:


   Apply transitivity of $B$ repeatedly.


More elaborately:


   If $w \in A^*$ then either $w = \varepsilon \in B$ or there are $w_1, \dots, w_k \in A \subseteq B$ with $w = w_1 \cdot \dots \cdot w_k$. Now, because $BB \subseteq B$ we have $w_2' = w_1 w_2 \in B$. For the same reason, $w_2' w_3 \in B$. And so on. Perform induction for the general proof.