Agüité comes from the Mexican slang verb agüitarse, which means "to get sad, bummed out, or discouraged". It's a very colloquial and authentic way to express feeling down, as if your spirits have been 'watered down' (related to agua).
In the song, the singer admits, "Si me agüité, fue porque al chile me importabas" (If I got bummed out, it was because I honestly cared about you). This moment of vulnerability, using such a casual term, makes the eventual triumphant chorus of "Qué bueno que te fuiste" (It's good that you left) feel even more powerful.
This song is the ultimate anthem for when you're finally over a bad breakup. Grupo Frontera isn't crying over a lost love here; they're celebrating it. The singer is talking directly to an ex who cheated, saying, "Qué bueno que te fuiste" (How great that you left). He's completely done with the drama, making it clear, "Ya no me pongo triste" (I don't get sad anymore). He's even erasing the memories by deleting old photos and refusing to dance when their special song comes on.
He admits it hurt at first because he genuinely cared, saying, "Al principio me dolió" (At first it hurt). He even calls himself out for being a "güey comprándote bolsas caras" (fool buying you expensive bags) while she was seeing someone else. But now? That pain has turned into pure relief. He ends the song by declaring they're enemies and that he's having a much better time without her: "Me la paso mejor sin ti" (I have a better time without you).