Why can an actor sit firmly on the horse after jumping off the horse?

The role of inertia.

When an actor jumps for a short time, he can ignore the air resistance. When he is in the air, the resultant force in the horizontal direction is zero, and his speed in the horizontal direction will remain unchanged, which is equal to his speed on the horse, that is, the running speed of the horse. As long as the horse's speed (including the direction) remains the same, he will not fall off the horse.