Saying "Tá uisce" just means "water is" so it wouldn't make any sense considering what is being communicated. There are two pieces of information being given here. The first is that "you have water/water is at you", the second is "you drink it". A simpler way of saying this would be "Tá uisce agat. Ólann tú é." Hope I helped! :)
Because, as you said, agat mean "at you." You are basically saying "you have water at you," which is the way to say you have water because of the way the sentence is structured.
You are basically saying "you have water at you,"
Except that, just to make it clear, there is no actual word for "
have" in the Irish. It's more "water
is at you".
Compare the Latin possessive construction "mihi est" ("it is to me"), Welsh (another Celtic language) "mae ... gen i" or "mae...gyda fi" ("...is with me") and the somewhat similar French "C'est à moi" ("It's mine").