Everyone has exactly 1 favorite book (z) I am learning discrete math and I fail to understand this:
Everyone has exactly 1 favorite book (z):
∀x(∃yF(x, y)∧∀z(F(x, y)→(y=z)))
I mean, this part I don't understand: ∀z(F(x, y)→(y=z))
Are there some good resources and examples to practice and learn it?

There is a typo in the formula …
It must be :

$∀x∃y(F(x,y) ∧ ∀z(F(x,z) → (y=z)))$.

The part $∀x∃yF(x, y)$ means : "for every (person) $x$ there is (at least) one (book) $y$ such that $F(x,y)$".
At least one means : "one or more".
Thus, to specify "exactly one" we have to add a formula that exludes the cases : two, three,etc.
This can be done saying that : "for every (book) $z$, if $F(x,y)$ then $z$ must be the same as $y$", i.e. :

$∀z(F(x,z) → (y=z))$.

