Two vector spaces $V$ and $W$ over a field $F$ are called isomorphic if there exists a bijective homomorphism of linear spaces $T : V \longrightarrow W$
Let $A$ and $B$ be two subspaces of $V$. Prove that $(A + B)/B$ is isomorphic to $A/(A \cap B)$.
I do not really understand the meaning of $(A + B)/B$ is it modular? If so, how do you divide by a set? same with $A/(A \cap B)$.
This is about quotient spaces. You'll find the definition on Wikipedia - Quotient space.
What you're requested to prove is Theorem B for modules in Wikipedia Isomorphism theorems.
For a proof, you can have a look at theorem 3.21 of this Quotient module paper. A vector space is a module!