Den Rest geben literally translates to "to give the rest". However, it's a powerful idiom that means 'to be the last straw' or 'to deliver the final blow' that emotionally breaks someone.
In the song, the line "Du gibst mir den Rest" conveys a deep sense of despair. The singer feels so overwhelmed by the painful breakup that this one final interaction is the thing that completely 'finishes him off'.
This song is about one of the most awkward moments in any breakup: giving back your ex's stuff. The singer meets his ex at a bus stop, and he's brutally honest about why they're even seeing each other. It hurts, he says, that the only reason they're meeting is "Weil bei mir so viel Zeug von dir rum liegt" (Because so much of your stuff is lying around my place). He can't stand seeing it anymore. The whole situation is a mess of mixed feelings. He sings, "Ich halt' dich nicht fest / Und lasse dich nicht los" (I don't hold you tight / And I don't let you go), which perfectly describes that feeling of being stuck after a relationship ends.
What makes the song so memorable is the chorus: "Es tut mir Leid Pocahontas" (I'm sorry Pocahontas). Calling his ex 'Pocahontas' is such a specific and personal detail. It must have been an affectionate nickname, and using it while he's apologizing makes the whole thing feel more painful and real. He even admits that staying together just because they've been a couple for a long time is a "verdammt beschissner Grund" (a damn crappy reason). It's that kind of blunt, sad honesty that makes this song hit so hard.