Definition of integers needs explaining
The natural numbers don't exist natively in the world of lambda calculus, and so if we want to use them, we have to define them somehow. You are describing the Church numerals, one possible definition which supports arithmetic. Church defines $n$ to be the function $(s,z) \mapsto s^{(n)}(z)$ that maps a pair $(s,z)$ into an $n$-fold application of $s$ to $z$. In particular, when $n = 0$ we get $(s,z) \mapsto z$.

Why this definition and not any other one? No reason. It is just a common definition, but you can use any other definition you like, or even come up with a new one. This particular definition supports arithmetic – you can define addition, multiplication and so on. See Wikipedia for the details.