Te metiste is a powerful phrase from the reflexive verb meterse, meaning 'to get into' or 'to put oneself in'. While it can mean physically entering a space, here it's used in a deeply romantic and figurative way.
The singer uses it to describe how someone completely and suddenly entered his life, heart, and thoughts. The line "Te metiste como el agua entre mis manos" (You got in like water through my hands) creates a beautiful image of how effortlessly and completely this person became a part of him.
This song is a beautiful declaration of a love that took the singer completely by surprise. He describes how this person entered his life almost without meaning to, saying, "Te metiste así como no queriendo" [You got in as if you didn't mean to]. But now, they are everything to him. He feels so much gratitude for the happiness they bring him, asking, "Como te pago todas las noches / Todas las veces que me has hecho tan feliz" [How can I repay you for all the nights, all the times you've made me so happy?].
His love is so deep that his own happiness depends on hers. He explains that he can't stand to see her sad, because "me pone a sufrir" [it makes me suffer]. The central idea is how this person has completely become a part of him, like in the line "Te metiste como el agua entre mis manos / Para quedarte siempre, siempre junto a mi" [You got in like water through my hands / To stay always, always with me]. It's a powerful image of a love that was unstoppable and is now a permanent part of his life.