First of all, that's a horrible sentence because everything besides "get stuck with" has a "formal" feel to it, and "get stuck with" is very casual.
Let me give you an easier example: "Chris was eating dinner with Joe. Chris (without Joe knowing) left the restaurant through the back door and Joe got stuck with the bill".
Harder example: "After the divorce, Ann got the house, but also got stuck with the mortgage and her husbands credit card debt".
So, I would say you "get stuck with" something if it's yours, you can't get out of the responsibility, and it's negative. Usually bills/debt/etc.