Regarding a theorem that says a language that is Turing-Recognizable iff some enumerator enumerates it
The Turing machine M is running the Turing machine E as its subroutine. In the other words, you can think of E's printer tape as another tape for Turing machine M. Although we don't know what is going on E, E's computation is part of (inside) M's computation. And of course, as nir shahr said in the comment, comparing is part of M's computation too.
I drew something maybe it can help to have some intuition.