The vestibular system is dedicated to your sense of balance and orientation in space. The structures of the vestibular system are located in the inner ear, with sensing neurons sending messages to various parts of the brain and spinal cord to coordinate body movement and maintaining balance. One of these structures is the semicircular canal, which is a pretty apt name for a half-circle shaped tube filled with fluid, specializing in detecting angular momentum-- spinning. The semicircular canals have a membrane on each end, and if the head rotates, the fluid in the canal, endolymph, gets pushed up against the that membrane. Think about if you were to put water in a Hula hoop, hold it horizontally, and then rotate the Hula hoop left and right. The water would pretty much stay still while the hoop moved around it; that's pretty much what's happening in your ear. Which membrane has the fluid press against it tells the brain which direction your head is turning.
Alcohol is a blood thinner (which is one reason why reputable tattoo and piercing parlors won't let you partake in their services if you have recently imbibed), but it also thins other bodily fluids, like, for example, endolymph. When you drink, the endolymph in that canal gets thinner and sloshes around more, even if you're not turning your head, right up against those membranes that tell your brain there's been angular moment. Your brain interprets that as "Weeee! I'm spinning!" and the spins are born.
We can take this one step further though, because why the spins are worse when you close your eyes is also all about the brain. When your eyes are open, you're also sending your brain visual messages that conflict with what your vestibular system is saying. Your ears say you're spinning, but your eyes say you're not. Your brain has to bring these two together, usually mostly trusts your eyes, and stands down a bit on the spinning interpretation. If you close your eyes, you no longer have any visual input to override what your semicircular canals are telling your brain, and off you go again into the land of swirly twirly gumdrops.