Before I begin, and just to make a point, while it was foolish to the Nth degree to change that switch with the breaker on, doing so has demonstrated a weakness in the wiring that you might not have learned about otherwise until a serious problem occurred. Do NOT take that to mean that it's ever a good idea to leave the breaker on. I'm merely pointing out that that a bad choice has had the unusual but not unheard of consequence of leading you to wisdom.
Don't depend on that.
The reality is that, bless his heart, your father made a mistake. It's human, we all get it. But it's better to not make the mistake. That's the wisdom you're learning here.
OK, the three most common reasons why this happens:
You have a metal outlet/switch box, which should be grounded, and while pushing the wires back something hot (usually the screw posts) touched the box. You're explaining damage to the ground wire and your box is likely modern plastic so this is probably not the problem.
You have a bare ground wire (very common) that bent over and touched a hot screw post (or a wire too long or otherwise badly connected to that post) on the switch (should be a mark on the post where ignition occurred). Note that if the switch is "on" then both posts are hot. If the switch was "off" only one post was hot.
You have a wire nut connecting two or more hot wires together and one or more of the wires was stripped too far, leaving a small portion of exposed wire outside the wire nut cap — and that bit of bare hot wire touched something like your bare ground wire or the metal mounting bracket of the switch (should be a mark on the hot wire near the base of the cap).
As I mentioned, I believe #1 can be ignored. It's caused by someone's shaking hands or a weird angle of entry before seating the screws that allowed the screw posts to touch the metal wall. Once installed, those posts and walls are intentionally separated and quite safe. However, that is only relevant if you have a metal switch box. Most switch boxes today are plastic to avoid this very problem (and a lot of others that are similar). However, once those screws are tight, the problem no longer exists — but you said you saw a mark on your ground wire, so it's not #1.
The other two demand that you remove the switch and check what happened because the problem still exists. That electrical short can happen again as wire tension relaxes or as your house shifts, which can gently move the wires. Or as use of the switch itself causes slight (very slight) vibrations inside the box. All of that might seem far-fetched... but it your house that's going to burn down with you and your family potentially in it. Are they so far-fetched that it's not worth re-seating the wires?
Check the wire nuts (white and black) to be absolutely sure no bare wire is peeking out from under the cap. In fact, you'd be surprised how often 3+ wires in a cap can result in one that slipped out a bit or is actually quite loose. This is notably true if you reused the original wire nut rather than using a new one.
Check the ground wire bundle to be sure it's pushed flat against the back of the box and check the wire from the bundle to your switch to be sure it has an appropriate "S" bend that will allow it to collapse away from the screw posts. Personally, I like to bend that wire to the top or bottom of the box (whichever is closer to the ground wire mount on the switch) and then S-bend it so that it's collapsing along the bottom or top of the box, not the sides.
While it's always good advice to invite a professional to review the work, this isn't something that's complex. You can do this. just remember to shut the darn breaker off. There is no convenience that's worth your life.
