What does the quotient group $(A+B)/B$ actually mean? I understand that $A+B$ is the set containing all elements of the form $a+b$, wit $a\in A, b\in B$. 
When you do the quotient group, that's like forming equivalence classes modulo $B$. All elements in $B$ should be congruent to 0 modulo $B$, as far as I understand. So wouldn't $(A+B)/B$ be the same as $A/B$?
Where is the error in my reasoning?

The error is if $B\not\subset A$ then we cannot take the quotient. This theorem essentially says if this is the case, then we can either expand $A$ to the smallest group containing $A$ and $B$ and then quotient by $B$, or we can restrict to $A$ and quotient by $A\cap B$ and, they give the same result.
Note that if $B\subset A$, then $A+B=A$ so the quotient is indeed $A/B$.
