There's more than one answer to your question, depending on how you look at it.
For instance, looked at psychologically, water is wet because that's how our brain is wired to interpret its interaction with our senses -- it's like asking why does concrete feel hard, or why does the sky appear blue?
Another way of looking at the same question involves water's molecular bonds. Water is wet because it bonds to lots of things -- including the molecules in skin, once the surface tension is broken.
Offhand, those are the only two ways I can think of to respond to your question. A psychologist or chemist would know more.