Whether or not the horse has a drinking problem is irrelevant to its essential nature. And even if it were touching its essential nature, such as whether it is, in fact, a horse, were questioned, the problem would not be "I don't think so", but "I do not think" in a generalized sense. And even then, it doesn't disprove existence, but rather, is incapable of affirmatively proving existence.

In short, it's not a philosophy joke. It's a joke by someone who doesn't understand philosophy, particularly Descartes' philosophy. I'm guessing whoever came up with it went to college, majored in communications, got Bs, drinks a lot of beer, watches football, and thought this would be a way to remind the rubes he drinks beer and watches football with that he did, in fact, go to college.