Given Newton's third law, why are things capable of moving?

I think it's a great question, and enjoyed it very much when I grappled with it myself.

Here's a picture of some of the forces in this scenario.$^\dagger$ The ones that are the same colour as each other are pairs of equal magnitude, opposite direction forces from Newton's third law. (W and R are of equal magnitude in opposite directions, but they're acting on the same object - that's Newton's first law in action.)

While $F_{matchbox}$ does press back on my finger with an equal magnitude to $F_{finger}$, it's no match for $F_{muscles}$ (even though I've not been to the gym in years).

At the matchbox, the forward force from my finger overcomes the friction force from the table. Each object has an imbalance of forces giving rise to acceleration leftwards.

The point of the diagram is to make clear that the third law makes matched pairs of forces that act on different objects. Equilibrium from Newton's first or second law is about the resultant force at a single object.

$\dagger$ (Sorry that the finger doesn't actually touch the matchbox in the diagram. If it had, I wouldn't have had space for the important safety notice on the matches. I wouldn't want any children to be harmed because of a misplaced force arrow. Come to think of it, the dagger on this footnote looks a bit sharp.)

I had similar problem in understanding the 3rd law. I found the answer myself while sitting in my study chair which has wheels!

sitting in the chair, I folded my legs up so that they are not in touch with ground. Now I pushed the wall with my hands. Of course, wall didn't move but my chair and I moved backward! why? because wall pushed me back and wheels could overcome the friction.

I was mixing up things earlier : trying to cancel the forces where one cannot.

Movement of the matchbox is due to the force which you apply on it. period.

Now why you didn't move when matchbox applied the equal force on you is because of the friction. If you reduce the friction like I did sitting in the chair, you would also move in opposite direction.

Equilibrium can only establish itself when the forces are on the same object..

Alas, I am free from this confusion.. such a relief

Forces related to Newton's third law apply to different bodies, therefore they cannot cancel each other out.

For example, the reaction to Earth's gravitational pull on the Moon is the Moon's pull on Earth. That force won't have any relevance to the Moon.