Where do the length restrictions of the pumping lemma come from?
The first condition, i.e. $|y| \geq 1$, is clearly necessary if you want to say something interesting: for $y = \varepsilon$, $xy^iz \in L$ trivially and always holds.

The second condition, i.e. $|xy| \leq p$, is "arbitrary": the lemma still says something interesting if you drop it, and it is still true because the statement becomes weaker.

But remember what we want to use the pumping lemmas for: we want to find a (sufficiently long) word such that all valid decompositions into $x,y,z$ fail to pump. Therefore, it is useful to allow as few such decompositions as possible. Lucky as we are, the proof of the pumping lemma readily yields a strong restriction, namely that there has to be a pumpable decomposition with $|xy| \leq p$ with constant $p$. 

Now we have to refute only finitely many prefixes $xy$ (with possibly infinitely many different continuations, of course). If you look at example applications, you will see that they make heavy use of this restriction.