How does the pump work?

There is a one-way valve in the pump.   The air is allowed to flow only in one direction.  When punched hard,  the one-way valve stops the air from going out of the valve but allows the air to enter the Flexi pipe after the punch.

When the rubber blower is punched hard, the air is forced out of the blower and goes into the narrow Flexi tube   The air is then forced into the straw.

Every action has an equal and opposite reaction. When the air enters the bent straw, the big puff of air gets stopped at the top and pushes back down into the rocket. The force of air pushing down launches the missile.