The true nature of T.H.E.M. and their involvement in the bigger plot is revealed in Chariots of the Dogs, where Sam & Max get into their UFO when they go after Bosco after his fears of being abducted by T.H.E.M. come true.
The background of T.H.E.M. is explained in The Mariachi Song. As the last in a line of mariachi's and one of the few of his people to care about birthdays, Pedro got the idea of using a time-traveling UFO to be able to be present at as many birthdays as possible. He used time travel to recruit two past selves to assist him, thereby becoming T.H.E.M..
However, time-traveling UFOs come at a big price, in this case a deal with Hell. As part of the agreement they use the UFO to collect souls (including those delivered by Jurgen) and transfer them to Hell through a bermuda triangle in the soul crushing room, after using the Moai Heads to process them.
At the end of the episode, the youngest Mariachi left with a record contract, the middle Mariachi was left in the 1960s and the oldest left the room and isn't seen. They then appear together in a single elevator before going to their previous appearances throughout the season. Their whereabouts after that are unknown. In The City That Dares Not Sleep, when Sam is in Max's stomach, sees a pair of Mariachi frogs and says "Max got me those Mariachi frogs for my birthday", he looks around, expecting one of the Mariachis to turn up, but none of them does.
In-game, the abbreviation T.H.E.M. is only used in sentences where "them" would be grammatically correct, so that for the first few episodes it seems doubtful that it is even a real abbreviation at all. Indeed, this view would seem to be confirmed by the line "If you explode it, T.H.E.Y. will come." in Bosco's back room. One possibility is that Bosco himself was unaware that T.H.E.M. is an abbreviation, and that there was thus a peculiar coincidence.