The sentence this corresponds with is "Some students in this class grew up in the same town as exactly one other student in this class." It seems that x and y are the same student from the same town and z is the 'exactly one other' student from that town, but is that wrong? Who or what is z? And why must x equal either y or z in the implication?
∃x∃y (x=y∧P(x,y)∧∀z(P(x,z)->(x=y∨x=z)))
The first step is to carefully define $P$. As best I can tell, $P(a,b)$ says $a$ and $b$ grew up in the same town. The issue is whether $P(a,a)$ is true. You could define $P(a,b)$ to require $a$ and $b$ be distinct or not. I think they have not required that. In this case, the first term in the conjunction should be $x \neq y$. If $P(a,a)$ were defined as false we would not need this. The first two terms now say that $x$ and $y$ are two different students who grew up in the same town.
Now we want to say that no other student grew up in the same town as $x$ and $y$. $z$ is a dummy variable for a hypothetical other student that grew up in the same town. I think a better way to say this is $\forall z((P(x,z)\wedge z \neq x) \implies z=y)$. Whoever wrote this was thinking that the two students could be reversed, but because the $x$ here is in the scope of the original quantifier it can't happen. We have already (with my correction) said $x \neq y$ so we don't need it again.