Why does the Pumping Lemma Constraint |xy| ≤ p mean that y can't be 1 in the string 0p1p
Let $p$ be the pumping length of your language.
As you say, the string $s$ can be written as $s = xyz$ where $|xy| \le p$.
By the choice of $s=0^p 1^p$, you know that the first $p$ characters of $s$ are all $0$, therefore $xy$ (which contains at most $p$ character) must be a string containing only $0$s.
Since $y$ is a suffix of $xy$, $y$ must also contain only $0$s.