This is a gerund phrase. When you need to make the subject of the gerund phrase clear, the most correct way is to use the possessive form e.g. their giving you .... However, using the object form (them, him, us, etc.) is also acceptable, though less formal. I don't know why that is the rule. Perhaps it is because we always expect a finite verb form after a subject (e.g. they ARE GIVING, they GIVE), not just a gerund.

"Them" is of course the object pronoun that is equal to = "to someone" Here the sentence is written from the point of view of the object (that acts as subject) Interesting!

That is about cases of pronouns. "the act of (you) giving something to someone and (of) them giving you something else" them=objective case they=nominative case of giving something to someone *of-preposition *("you" which is omitted) [...]-object of the preposition which is in objective case *and-conjunction *(of) them [...]-object of the preposition which should also be in objective case. Please see more info on cases of pronouns to better understand. ☺