Step-by-step explanation:
this is almost similar to how a syringe works. when the pump is drown out, air in the barrel expands and pressure is reduced to below atmospheric pressure. the air from outside the pump is at high pressure so it flows past the leather washer into the barrel. at some point, the higher pressure in the tyre closes for valve.
when the pump handle is pushed in the air is forced into the tyre though the tyre valve which now opens