A top-out is essentially a way to go from a hanging position on a ledge (a "cat"), to having your upper body above the ledge with your palms supporting you, without clambering up with your elbow in between. Here's a mock up of it:
And a video (if you only want to watch the top-out, and not all the instruction, you can go to 5:12):
It relies on three things: a solid footing, a good knee drive with the hanging leg, and of course maintaining a solid grip with your arms. When done properly, it requires a lot less upper body strength then you might imagine.
For a brief overview, it consists of three parts: building upwards momentum with your legs, building a bit of forward (but mainly rotational) momentum with your arms (the reason why they play more into rotation more than anything will be discussed), and finally transitioning to the support position resting above your palms. First, and most important, is the legs. One is planted firmly, and the other is supposed to drive upwards, in order to build momentum which will later be transferred to the rest of your body. However, friction can be tricky: the tendency of your planted leg is to slip and slip, because most people will "paw" at the wall as if they were running up it. As we know, frictional force is proportional to normal force, so you actually want to kick/jab your foot into the wall, because this will allow it to stay in place. As you're doing this, you can drive your hanging leg up, generating some momentum.
During this, you should also be pulling up/in with your hands. Simply, you want to counteract/overcome the force of your leg pushing away from the ledge, and also gather a bit more upwards momentum. However, simply due to the weaker nature of our arms, it won't contribute quite as much as our legs, which can be surprising. What is helpful, though, is the torque it creates on the body: while it is counteracting the linear momentum from our legs, it is working with the force from our legs to rotate our body over the lip, which is more beneficially, seeing as we right next to the wall to begin with. Now, with momentum built up from our knee drive and arms, and a slight rotation, our upper body will pop up and over the ledge, and rotate us into a position where we can easily re-position our hands to rest on our palms.
From there, it is usually pretty easy to swing/climb up the rest of the way. But without proper training, this technique is very difficult, because people usually rely on their arms way too much. Yet again, it's an example of something made easier through physics.