Let’s try a different approach to explaining an engine stalling. Say you have a regular lawnmower. A rotary blade with a small gas motor. The motor has a shaft the spins and on this shaft the blade is attached. There is no transmission, no belts, the blade is directly attached to the motor.
You come home from a month vacation and instead of the normal 2 or 3 inches of grass you need to cut, it is over a foot high. You fire up the lawn mower and move it into the high grass and it stalls, the motor stops running. You pull it out and start it and try again, same result. Next time you start the motor, push it in the grass and you hear the motor straining and just before it quits you pull the mower back out a bit and the motor keeps running and gets back up to speed. You find out if you go real slow and work the mower back and forth you can keep it running. It takes a long time but eventually you mow your hay field of a lawn and head inside for a beer.
Your actions in pushing the mower into and out of the high grass to keep the motor running is essentially what you are doing with a clutch. A gas motor that is overwhelmed by the load on it will stall. You are using the clutch to control the “load” on the motor to keep the motor from stalling and get the car moving.