Why is the pressure inside the bottle lower than atm pressure? When a balloon is placed inside a bottle (covering the entranace), it will not inflate, since the bottle is already filled with air particles with no escape route. When you punch a hole in a bottle, the air molecules in the bottle have an exit. They are pushed out as a balloon fills the space inside, resulting in room for the balloon to inflate.
If the hole in the bottle is then plugged, the balloon stays inflated even when the mouth is removed. This is because the high pressure air in the balloon pushes outward harder than the low pressure air in the bottle. But, why is the pressure inside the bottle lower? There is less air, which makes the pressure lower, but there is also volune reduction, as the balloon takes up part of the space of the volume, which makes the pressure higher. Why is the first factor higher than the latter?
 A: In fact, just before you plug the hole the pressure of the air inside the bottle is the same as the atmospheric pressure, as you said that there is less air but also less volume. When you plug the hole there are three forces: the elastic force of the balloon to make it shrink and be smaller, the pressure from the air outside the bottle, and the pressure from the atmosphere that wants to inflate the balloon. When t=0, which means that you have just plugged the hole, both atmospheric pressure and the pressure inside the bottle are equal, so the only force that acts is the elastic one. But as the balloon shrinks a little bit, the air inside the bottle has less pressure as it has the same number of molecules but more volume, and the atmospheric pressure stays the same (don't think about the volume that loses the room, it is negligible, just imagine you are outdoors if you wish). Then the pressure of the atmosphere is stronger than the pressure of the air inside the bottle, counteracting the elastic force and avoiding it to shrink.
Note that, the balloon has to shrink a little bit, but sometimes this difference of volume is difficult to see.
