"To come back to Earth...it can be five times the force of gravity" - video editor's mistake?
Reentry speeds are fast. Astonishingly fast. The shuttle reentered at 7.8km/s. Now note the units. That's "per second." That's 28,158km/hr. And you have roughly 100 vertical kilometers to do that braking in. Yes, the braking gets to be done at a very shallow angle, which means you have more linear distance to break than the 100km would suggest, but its still a very short time to lose a ton of speed!
This requires a pretty substantial braking force. The force they are referring to is the aerodynamic forces felt by the airframe as it starts to bite into the ever thickening atmosphere. The steeper one's reentry, the more the force has to be.
This force causes a deceleration, of course. And it is not easy for humans to come to grasp with 50m/s^2. It just isn't a concept we have a good intuitive sense of. So what we tend to do is phrase it in terms of gravitational accelerations, dividing out 9.8m/s^2. We can intuitively grasp the idea of feeling like you weigh five times as much as you do when you are standing upright.
We endure these brutal reentry forces, of course, because there's a balance to be played. It would be possible to reenter slower by taking a more shallow angle. However, it can be tricky to control in this environment, and you run into additional heating problems because you spent more time at temperature as you slowly dropped your speed.
This is the "braking acceleration" from atmospheric resistance that slows down the descent. The capsule is specifically shaped to provide sufficient but not too great deceleration. See the article Returning from Space: Re-Entry on the web.
From a general relativity point of view, no objects, free falling or otherwise, are subject to a force of gravity, as gravity isn't a force, it's a manifestation of the curvature of space time. But to the extent that there is such a thing as gravitational force, objects in free fall are subject to it just as much as other objects.
Now, the way we generally experience gravity is not directly. Rather, we experience the normal force between us and the ground that results from and resists gravity. A person in free fall won't feel gravity, in that there will be no normal force resisting gravity, and so they won't feel anything pushing on them, but they are accelerating with respect to the Earth's reference frame, and so in that sense are undergoing a force.
If someone undergoing re-entry is subject to a braking force that causes them to accelerate five times as much as Earth's surface gravitational acceleration, then they are subject to a force that is five times the normal force that would be required to resist gravity. Since the normal force is equal in magnitude (though opposite in direction) to gravity, this is equivalent to saying that they are subject to a force five times the gravitational force.
It might be a bit more accurate to say that they are subject to five times gravitational acceleration, rather than force, as the former is an intensive property and the latter is extensive, but the latter is still a reasonable thing to say.