It means: if I push against you, you are also pushing against me. A book on a table is held up by the table, and so it must also push down on the table. If a bullet is pushed forward by a gun then the gun must also be pushed back, which we call recoil.

In (Newtonian) physics, we think of things in terms of objects (which we assume are rigid, that is, they're like hard balls on a pool table and never squish or heat up or anything), forces, and position/motion.

Each object has, at any given moment, a position. That position is changing over time, and the change is called the object's velocity. And, in turn, the change in the object's velocity is called acceleration.

Changing the motion of a heavy object is harder than changing the motion of a light object. This is where we get the ideas of momentum (the velocity of an object times its mass) and of force (the acceleration of an object times its mass). Or, equivalently, you can think of force as the change in momentum over time, in the same way that acceleration is the change in velocity over time.

In the way I'm presenting it here, the definition of force gives you Newton's second law (the equation F = ma, that is, force equals mass times acceleration) for free: that's just what we mean by the word "force". It also gives you the first law: if force is zero, acceleration is zero, and therefore change in velocity is zero, too.

Newton's third law, on the other hand, says that all the forces in a system (possibly on more than one object) always add up to zero. In other words, if I apply a force +F to you, there is necessarily a force -F applied to me. If you're pushed forward, I'm pushed back. If you're pushed up, I'm pushed down. And so on.

But remember how we said earlier that one way to think of force is the change in momentum? Well, if all the forces in a system add up to zero, the total change in momentum must, therefore, also add up to zero. If you gain momentum upward, I must gain momentum downward; if you gain momentum to the right, I must also gain momentum to the left. So a modern way to describe Newton's third law is just that it describes conservation of momentum: you can never create or destroy momentum, only transfer it between objects.

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What happens when you jump? You flex your legs and push the ground really hard. Eventually, you get away from the ground.

But why? You pushed DOWN, against the ground. How come you're going UP? That's Newton's 3rd law. As you're pushing the ground, it is also pushing against you. Since usually the ground is much heavier than you, it barely move if at all, while you move a lot more.

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.)

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Just remember, the force never just goes away, it doesn't disappear into nothingness. It has to go somewhere. So if I chuck a bunch of mass behind me, like if I am carrying fuel and I burn it and it leaves me by escaping as focused gasses, where does the force of the escaping molecules end up? Intuitively you probably know this, the top of the combustion chamber.

Have you every stood up in a canoe or similar boat and picked up a cooler and handed it to someone on the dock? What happens to the boat? It starts moving away from the dock a little bit. As I hand it to the person on the dock the mass that was in the boat has moved and as a result the force of my moving it is transmitted through my legs and into the bottom of the boat. The boat moves in the opposite direction of where I moved the object of mass.

The why to all of this is a deep discussion about gravity, mass, and the nature of time.

Basically, any energy has to go somewhere. If you hit a billiard ball into another, the second ball will travel away from the first, and the total energy after the hit will be about the same as before (minus friction and other losses). Pushing someone forward will tend to push you backward, etc.

If you thinking of one of those swinging ball toys, Newton’s Cradle, this law is what you see when you pull back one ball and let it go. The action (one ball swinging into the group) has an equal and opposite reaction (one ball swinging out of the group). Pull back two balls and let them go, and now two balls swing out the other side.

Hope that helps.