What physically happens when a person sleeps with their hands and arms tucked under them?

Nerve entrapment. If you sleep with arms and hands tucked under, that sleeping position can cause compression or entrapment of your ulnar nerve at the elbow, known as cubital tunnel syndrome, resulting in numbness in the small and ring fingers. The numbness is usually transient, but can become chronic if the nerve entrapment continues.
Sleeping on arms. A variety of things happens depending on the exact position. Occlusive pressure on nerves or vessels can produce symptoms. One example is polularly called saturday night palsy when someone passes out with their arm over a chair. Stress on the joints can produce other symptoms releated to nerve compression or stress producing weakness and/or numbness.