Where does the force of air pushing on something come from?

Before the box is opened, air molecules are bouncing off of every surface of the box. Let's focus on the surface with the door (call it the "front" wall), and the opposite surface (the "back" wall). Molecules bouncing off the front wall exert a force on the box in the forward direction, but these forces are balanced by molecules bouncing off of the back wall.

When the door is opened, molecules simply escape through the door and no longer bounce off the front wall. Therefore the forces due to molecular collisions on the back wall are not compensated by collisions with the front wall, and there is a net backwards force on the box.

As Andrew’s answer says, when you make a hole in the wall of the box it does not create a new force - it is the removal of the pressure on this part of the wall that makes the forces on the box unbalanced and so the box moved in the opposite direction.

Another way to think of this is that conservation of momentum tells us that the centre of mass of the box plus the air that it contained cannot accelerate (since there are no external forces). But as the air escapes in one direction its own centre of mass accelerates, so the box must accelerate in the opposite direction to conserve momentum.