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

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.

As the box is opened,the air inside will rush outside and the box will move in opposite direction because of Newton's third law of motion

Andrew and gandalf61 posted some great answers, but neither (directly) addressed Newton’s third law (gandalf61 did mention conservation of momentum, which is related).

Newton’s third law says that whenever an object A exerts a force $mathbf{F}$ on an object B, B exerts a force $-mathbf{F}$ on A. That is, Newton’s third law is about pairs of forces. But none of the posts so far mention any pairs of forces.

Let’s introduce one. But we won’t look at air molecules, because they are too difficult to study individually. Instead, we can consider all the air molecules collectively as a single object. The other object is the open box.

As the other answers have established, the air pushes on the back of the box. This means that the box is pushing on the air, in the opposite direction – towards the front – with the same magnitude of force. If the air is lighter, then it will have a greater magnitude of acceleration, which results in a greater magnitude of velocity, which balances with the lower mass to result in the same magnitude of momentum. If the air is heavier, the same ideas apply, with everything reversed.

The previous paragraph shows how Newton’s third law is related to conservation of momentum, as mentioned in gandalf61’s answer.

It should also show how Newton’s third law, by itself, doesn’t really explain anything. Phrases like the one in your question suggest that the writer misunderstands this law; hopefully this answer will clear that up.

The conservation of momentum (as discussed in answers by others) applies considering the entire fixed mass consisting of the box and the mass of air moving into (or out of) the box. Considering the box and its contained air alone, you need to be careful as this a system of variable mass if the mass of air inside the box changes. The Halliday and Resnick Physics tests provide a simple discussion for systems of variable mass.

A rocket is another example of a system with variable mass.

From a thermodynamics point of view, a system with variable mass is an "open" thermodynamic system (one with mass transfer in/out). See one of the Sontagg and Van Wylen textbooks on thermodynamics.

What is pushing the box is the rushing air.

If you somehow removed the whole box at one moment the air would still rush, but what would there be to push?

If you pierced all six sides of the box at once, you might see the same result.