What is the importance of the condition "| xy | < p" in pumping lemma?
$p$ is the number of states in the automaton.  As you say, the pumping lemma is about the pigeonhole principle.  Suppose that $q$ is the first state that's repeated when you read input&nbsp;$w$.  Then $x$ is the string that you read before you reach $q$ for the first time, and $y$ is the string you read between the first and second visits to&nbsp;$q$. Since there are only $p$ states including the start state, you must repeat a state after doing at most&nbsp;$p$ transitions, i.e., reading a string of length at most&nbsp;$p$.  So the condition $|xy|\leq p$ comes from the proof.

In reality, when an automaton reads a particular string&nbsp;$w$, there might be many times when it returns to a state that has already been visited. If you take $w\in L$ any split $w=xyz$ with $|y|\geq 1$, such that the automaton is in the same state at the beginning and end of&nbsp;$y$ then $xy^iz$ will be in&nbsp;$L$ for all&nbsp;$i$, regardless of how long $xy$ is.