We also know that $x$ and $y$ are positive integers. I understand that if $x+y$ is prime, we know that it must divide either $x$ or $y$. But when it isn't, I can't find any other ways to find properties of these numbers.
If the given is true, then $N \in \mathbb{Z}, N(x+y) = xy$ should also be true. As far as I can tell, either $x$ or $y$ must be a multiple of $N$ and $x+y$ can't be coprime to one of $x$ or $y$, and maybe both. The problem is that I'm not sure if this is true in all cases and can't think of ways to prove it.
Thanks for your help!
Using @BarryCipra's solution we proceed as follows.
Note first that if $a,b,d$ are any positive integers, then $$x=da(a+b),y=db(a+b)\quad (*)$$ gives a solution, because $x+y=d(a+b)^2$ divides $d^2ab(a+b)^2=xy$
Now suppose $x.y$ are any positive integers satisfying the condition that $x+y|xy$. Let $x=ga,y=gb$ where $a,b$ are coprime. Then $xy=g^2ab,x+y=g(a+b)$, so $x+y|xy$ implies that $a+b|gab$. But $a+b,ab$ must be relatively prime (since $a,b$ are), so we have $g=d(a+b)$ for some $d$. Hence $$x=da(a+b),y=da(a+b)$$ So (*) gives us all solutions.