-p wraps a loop around your code. You can see what it looks like:

$ perl -MO=Deparse -pe '$\=$_}{'
LINE: while (defined($_ = <ARGV>)) {
    $\ = $_;
}
{
    ();
}
continue {
    die "-p destination: $!\n" unless print $_;
}
[download]

In effect, the }{ breaks off the continue part of the while loop that usually prints every line, so it only runs once at the end of file. Since $_ will have been set to undef at end of file, copying it to $\ preserves it so that print will show it. ($\ is implicitly added to each print by perl.)