When you lean against a wall, you apply force against the wall and yet the wall does not move.

But when you lean against a chair, the chair moves.

What is the correct scientific explanation for it ?

Newton's Laws apply

Let's say you push against wall with force 10 N. This is action. As a consequence of Newton's Third Law of Motion, the wall applies equal (10 N) amount of force on you. This is reaction. If you further increase the force to, say, 100 N, the wall applies the same amount of force on you.

As long as the two forces are equal, none of the bodies will move. For one of the body, say, the wall, to move, the force applied by you should be greater enough than the reaction force that will be applied by the wall.

In case of pushing against a chair, the chair is incapable of providing as much reaction force to balance the action. As a result, the action dominates and hence, the chair moves.