Aventar literally means 'to throw', but in this song, it's used as a popular slang phrase meaning 'to go for it' or 'to take the plunge'.
The singer is hesitating to approach a girl at a party, but then decides, "mejor me voy a aventar" (I'd better just go for it). This phrase perfectly captures that exciting, nerve-wracking moment of taking a social risk, which kicks off the whole story of the song.
This song tells a whole story from start to finish. It begins with our narrator at a club, setting the scene with lines like "una bolsita para dar el levantón" (a little bag for a pick-me-up) and getting ready to party. He's just hanging out when he spots a girl and is immediately floored. He says his heart starts pounding—"Llevaba tiempo sin eso experimentar" (It had been a while since I'd experienced that)—and he thinks he might be feeling butterflies.
He works up the nerve to talk to her, and when he does, she pulls him onto the dance floor. He's completely captivated, admitting "Tengo yo miedo y no me quiero enamorar / Pero ya es tarde y créanme que esto me gusta" (I'm scared and I don't want to fall in love / But it's too late and believe me, I like this). They spend the night together, but when the sun comes up, she's gone. The rest of the song is about him waiting for her, hoping to see her again. He's back at the same club, looking for her in the crowd and asking the moon to tell her to come back: "Dile que vuelva y aquí la voy a esperar" (Tell her to come back, I'll wait for her here).