Popped lamp fixture/socket?

The wiring is probably still OK. You blew out your dimmer switch, rather than tripping a breaker. You will need a new dimmer switch after you've repaired the light socket. Turn off the breaker before you do anything else.

Not a very handy sort of chap are you, and i suspect that you know very little about the workings of such a fitting. To save yourself much frustration, probably making the situation far worse than it is already, with the possibility of a mild case of electrocution, I suggest that you call out a professional electrician as soon as possible. If you don't know anything about it, don't bugger about with it, its dangerous, and thats what the professional are for.

the only thing to do is turn the power off at the breaker box and disassemble/inspect each component. test them with your multi-meter as you go. reassemble only those parts known to be good.

First, unplug the lamp if you haven't done so already.

If you haven't already unplugged the lamp, maybe the rest of this whole project isn't for you.

Reach in there with folded-many-times paper towel to grab whatever bits you can and try to unscrew the bulb's base. Paper towel so it won't be too rigid and cause any glass bits to break, folded many times to protect you from sharp edges.

If you have to, use needle nose pliers to grab the edge of the bulb base - being careful not to also grab the threaded metal part of the socket - deform it a bit by twisting counterclockwise, while also trying to turn the base counterclockwise. I have had some luck doing this.

If you wind up destroying the socket, new ones are cheap, just a few bucks. There are several useful video tutorials on YouTube to walk you through the process of replacing one. Watch it, pause, rewind as many times as you need to to understand what you're doing.

The short circuit may have cooked the dimmer, so you might wind up replacing that too.

Good luck with it.