If you have array @foo in perl, $#foo is the index of the last element of the array, which is just the size of the array -1. So if @foo is undefined, $#foo is -1.
Using a variable instead of 'foo' is a symbolic reference, so this is effectively using the symbol table as the associative array. This means that this solution also gets it wrong if your file contains a line that matches the name of a built-in variable in perl. That would be tough to debug!
If your file contains
This is the first line of the file
then during execution of
++$#$_
the result is the same as if you had written
++$#{This is the first line of the file}
So the variable @{This is the first line of the file} goes from undefined to an array of length 1, turning $#{This is the first line of the file} to 0.
Incidentally, this is why the snippet fails to work for a line repeated more than once: for each occurrence of the expression, the value returned is in the sequence -1, 0, 1, 2, 3, ... so it is only false for the second occurrence.
Using preincrement instead of postincrement means the values returned are 0, 1, 2, 3, ... which means that inverting the test makes it false for every occurrence after the first.
