What makes a tornado spin

Tornadoes form when warm, humid air collides with cold, dry air ... The updraft will begin to rotate if winds vary sharply in speed or direction. As the rotating updraft, called a mesocycle, draws in more warm air from the moving thunderstorm, its rotation speed increases

The intense spinning of a tornado is partly the result of the updrafts and downdrafts in the thunderstorm (caused by the unstable air) interacting with the wind shear, resulting in a tilting of the wind shear to form an upright tornado vortex.