That's a tantalum timebomb that went off alright. You probably didn't do anything wrong, they just fail after a couple of decades - and do so pretty dramatically.
Replacing the cap is trivial - although I'd recommend replacing all of them because if one failed the rest probably will soon too.
The resistor is more worrying. It's almost certainly the consequence of the short caused by the tantalum cap failure. If that resistor was all that failed, you can just replace it, but by the time resistors get so hot as to cause that kind of damage, a lot of other stuff might also have failed. First up a check: the damage appears next to the leftmost of those three resistors. Are you sure it came out of the resistor and didn't burn through the PCB from below?