English:
People pay this man so that he can solve the problems they have with their cars.
Deutsch:
Leute bezahlen diesen Mann, damit er die Probleme, die sie mit ihren Autos haben, löst.
What does "die" mean in this sentence?
If it means "that", then why don't we use "das" or "dass" instead?
It is a relative pronoun that refers back to die Probleme (hence die, and not das). You could replace it with welche without changing the meaning. It is the same as the following English equivalent:
People pay this man so that he can solve the problems, which they have with their cars.
It is die (and not das) because the accusative plural of das Problem is die Probleme. And it can never be dass, because dass never introduces a relative clause.
German doesn't offer the possibility to omit the relative pronoun the way English does in your example sentence.
