and just to prove how dumb I really am... I don't understand them.
Not really--they weren't exactly user-friendly instructions. In particular, he's assuming that you have already arranged to get shell access on your TiVo. Which is, alas, not a trivial task. See, e.g., www.tivofaq.com/hack/faq.html FAQ #4.6 for more info.

How do I go about uploading the script to my TiVo?
If you're using something like HyperTerm to access the TiVo, you could just "cat > padhack.tcl" and then send the file.

Where he references /var/hack, I am guessing he means create a directory called /var/hack (if it doesn't already exist) and somehow put the padhack.tcl script there.
Yes, though it doesn't actually make any difference where you put it.

chmod 777 probably means set the permissions to make padhack.tcl executable?
Yes.

If that's the case, ./padhack.tcl would only run if /var/hack were my current active working directory, right? So I would need a cd /var/hack somewhere along the way.
Yes.

If I try this (assuming I figure out how to load the script into my TiVo) is there a chance I could mess up my TiVo and make it not work any longer?
Yes.
If I did that, is there a way to fix it?
Restore your backup. You did make a backup didn't you?

Truth is, it's probably not a good first-time TiVo hack. The instructions are vague and assume some prior hacking as well as a bit of *nix familiarity.

--John