When the pump is switched on and the air in the jar is gradually removed, the pressure in the jar decreases. There will be fewer air molecules per unit volume in the far. Hence rate of collision of the air molecules with one another and with the wall and hulk will be reduced. As pressure P = F/A, the force acting per unit area decreases, the pressure decreases.
In the marshmallow, there are pockets of air at normal atmospheric pressure initially. As the pressure in the jar decreases, the pockets of air in the marshmallow expands due to this pressure difference. Hence the hulk expands and its volume increases.
Glad to see such projects which benefit the environment.