Your sense of balance is maintained (in part) by the "vestibular apparatus".
Within the vestibular apparatus are three fluid-filled tubes called "semicircular canals".
They're oriented in three planes, and gravity causes the fluid in them to shift. The differences in the fluid levels in each tube tell you how you're oriented with respect to gravity.
But spinning around rapidly causes the fluid in the tubes to spin. It remains spinning for a bit even after you stop moving. This confuses your brain, which thinks that the world is still spinning, even though you obviously aren't.
Your brain interprets that sensation as "dizziness".