The neurons in your spinal cord are partly responsible for this. Ever had a foot fall asleep after sitting on it for too long? This is because the signal wasn’t getting to your spinal cord. Once you let that signal have a proper pathway, you get the familiar pins and needles feeling.
The other neurons that are responsible for this feeling are located in your limbs. The sensory pathways that carry that kind of feeling to the brain extends down into your fingers and toes and comes back to the brain. If you pinch an area long enough - by sitting on it or putting enough pressure on it that the signal can’t get through - you’re blocking part of that sensory pathway.