+($\=\1)-$\ evaluates to zero, so we can take those out.
+($/=\1)-$/ evaluates to zero too, just we need one of them to set the input separator $/ to 1 (bytewise reading).
We can also take out the subsequent commas from the join (since they do nothing), so the first line becomes:

open OUTPUT, join "", map chr,112,114,116,51,46,116,120,116;
$\="";
$/=\1;
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The numbers are the ordinals of the single chars of the filename "part3.txt".

Now for the while loop. If we write it clearer it looks like

while(<OUTPUT>)
{
    $\="$\$_";
    $_=$\;
    $\ = substr((push @w,$\),0,0) if $\=~ /\s$/msg;
}
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$\="$\$_"; simply concatenates the read char with $\.
$_=$\; has no effect.
$\ = substr((push @w,$\),0,0) if $\=~ /\s$/msg; pushes $\ to an array if the last char of $\ is a whitespace or newline, thus a "word".
In the same statement $\ is cleared because the return value of substr(something, 0,0) is empty.

The last part is a simple grep of the array we built. We can safely leave out the eval so it becomes:

print grep{ /^\b.*(p).*$/msgi } @w
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Alter the regex here to your liking.