Let's say you want to carry a box. Does the box feel heavy? That's because it's weight is exerted on you. You have to carry its weight in order to lift it. But the box is also being lifted because you are exerting a force on it. This happens always, in all scenarios and circumstances.

Let me give a few more examples that are easy to see, but you might not have thought about them. If you stick your hand out a car window, you feel a force pushing it backwards. The force of the air slamming into your hand. But also realize that if your hand weren't there, the molecules of air would be relatively undisturbed. You are pushing them out of the way to make room for your hand, so you are applying a force to the air molecules as well.

A rocket works because the burnt fuel is forced down and out through the nozzle by the expanding gasses behind it. The expanding gasses apply a force to the spent fuel, but the spent fuel also pushes back on the rocket, providing lift. (If you wanted to get right down the the bottom of it, the spent fuel is pushing on the expanding gasses of the currently burning fuel, and that causes them to push on the rocket because they want to go up, however, the body of the rocket gets in its way, pushing down on them to keep them contained instead of moving up... Just another example of reaction force)

Walk up to a wall and push on it. It might be easier to see with something very heavy, but still just barely movable. If you don't brace yourself, when you push on it, you will end up moving backwards instead. That's because it's pushing on you with the same force you are on it. If you do brace yourself, you might be able to move it a bit. You'll notice that you feel in your ankles that you are still being pushed backwards, and into the floor, meaning your legs are also pushing on the floor with a force, but the floor is pushing back on you (with friction) keeping you from moving backwards.

Every. Single. Force has an opposing force to consider. The fact that it is exactly the same magnitude I don't know if I have the ability to understand well enough to eli5, but the opposite reaction clearly makes sense to me. If I had to try to phrase what I think causes the equality part, I think it would involve the conservation of energy & momentum. I think if you pushes on something and it didn't push the same amount back on you, then you would give it more energy than you lost. I think that is the main reason. And since energy can't be created or destroyed, the forces have to be equal. (Big disclaimer: this last bit is speculation, not a guaranteed correct answer.)