"What exactly makes the matter “want” to go into a empty space? What creates that pressure? "
My understanding is only general, but I don’t think that it’s the case that matter wants to go to a vacuum, so much as that when there’s an imbalance of pressure (in this case between a pressured space and a vacuum), the pressure from the pressurised space has nothing to counter it, so matter is pushed in that direction.
Just as in a balloon the air is under more pressure than the air outside it - once there’s a way for it to escape, it just will do, until the pressure is equalised.