When playing kings on a low flop board without much draws but an aggressive opponent. How do you know when they have a set or just a lower over pair? If the pot looks like it will be reaching all in by the river should you just call it down? I feel like I run into sets more often than i run into jacks or queens. This is for lower stakes as well.

How do you know when they have a set or just a lower over pair?

The short answer is you don't, this is why you have to play ur opponents range and not his specific holdings in these spots, they cant hit two pair or better every time and for this reason, in cash games, u should be willing to stack off when playing a particularly aggro opponent, its a messy battle but in the long term you'll beat these player types by trusting in the mathematics of the game and knowing they cant have it every time, try playing bk at them every now and then and see how they react to ur resistance, look out for sizing tells to help with ur decisions, in weaker games often the bet size will tell u weather they have it or not, ask urself if they had a set on a dry board would they really just be blasting away or would they be looking for value?

Hope this helps, GL

Are they a good aggressive player or bad aggressive player? Did you give a decent indication pre that you were holding a monster? Especially easy for them if you are otherwise tight/passive.
A good aggressive player:
Knows that you think they are aggressive
Often knows when you have a premium pocket holding
Will unlikely get it all in deep with just top pair

One of the main purposes of repeated aggressive play is specifically to exploit these spots where you put down a massive reraise just to their aggression, you have many hands beat, you follow them all-in, and they have you crushed.

You can wait til you're the one that flopped the monster, they have just top pair and they attribute your playing-back to your assumption that they're over-aggressive, and then you're the one collecting all the chips.
Or you can just join in the fun and look like you're aggressively over-adjusting, and then your KK more likely just to be up against a tptk jam.
Routine aggressive playback against a skilled aggressive opponent not for the faint-hearted.
If they're really really good you might just be jamming pre ����
Then they have Aces ��