Here's a breakdown:
* gratias: thanks, gratitude
* tibi: to you
* ago: I give, I offer
So, the full phrase literally translates to "I give thanks to you."
Other Latin phrases for "thank you":
* "gratias agimus" (plural, "we thank you")
* "gratias tibi ago maximas" (more formal, "I thank you very much")
* "multas tibi gratias" ("many thanks to you")
It's important to note that while "gratias tibi ago" is the most common phrase, other variations may be used depending on the context and formality of the situation.