Due to a liquid inside and organ in the ear. The liquid moves around when we move. Tiny hairs detect the movement and allow us to sense our exact position, a kind of internal gyroscope. The brain then takes that information and combines it with the info it gets from your visual cortex. This communication can be affected by other chemicals like drugs and alcohol. Notice that you really get dizzy when you stop spinning. This is because visually, you have stopped moving, but the liquid is STILL moving. Thus your brain is receiving conflicting signals which causes dizziness. With other chemicals, the electrochemical conductivity of the brain is altered and the signals are no longer synchronized. The effect is the same. People fall down or get into car accidents.
Your eyes. and yes, the ear thing. Your eyes try to register what is going on around you. As you spin, your eyes adjust to the new surroundings, and your ears have that fluid in it so pressure builds up and you stagger around trying to stop the madness :D
Because there is a part of the ear called the semicircular canal that has liquid in it. When you spin the momentum keeps the liquid moving, and like Virrey said, the hairs send information to our brain. This is also why some deaf people might not have good balance. (we had a whole unit on the ear in my ASL class)
The semicircular canal connected to your eyes that tells your brain what way you're facing, it is full of liquid. when you spin the liquid sloshes around and keeps moving after you stop. This tells your brain that you are still spinning and creates that dizzy feeling.
Our interior balance is solely dependent on our eardrums. As our head is filled with blood, spinning will disrupt the balance by taking some pressure off one ear and applying it to the other ear. Though not major damage, this sudden effect really freaks your gyroscope out, causing your lack of balance. A similar effect takes place during and after a blood rush (holding your head upside-down for too long and allowing blood from the rest of your body to drain into your head. If in this state for too long, death is possible).