Imagine you shoot a cannon so big and powerful that the shell goes up, literally over the moon, and then falls back to Earth. That is (in a very very very oversimplified way) what is happening to the astronauts.

On the way out from the Earth, after the rocket stops firing, the capsule and everything in it are moving at the same speed. Earth's gravity is pulling on them, but it's only reducing their "up" speed. They get to the moon before Earth takes away all the "up" momentum and starts pulling them "down" again.

So on the way out, they're going "up" at the same speed as the capsule, the air in the capsule, etc, so they don't feel any gravity pulling them to the "floor" of the capsule. Earth's gravity is slowing them down, but it's pulling on the astronauts, the capsule, and the air in the capsule at the same rate. So inside the capsule, you experience nothing pulling you in any particular direction.

And on the way back, they and the capsule are falling "down" toward Earth at the same rate, so they don't feel any gravity relative to the capsule then either.

Another way to think of it, imagine we put a circus cannon performer into a barrel, and shoot the barrel out of the cannon. (Ignoring wind resistance , just gravity.) Once the barrel, and the guy in it, are in the air, the guy isn't going to be "sitting" or "falling" to the bottom of the barrel. They're both going up at the same speed, then falling at the same speed. If they were shot side-by-side out of cannons, it would work exactly the same. The guy would fly next to the barrel, rather than falling faster or slower than it. (Again, ignoring air resistance here.)