The short answer is, they don't. What I think of when you say anti-gravity is the sort of levitating spaceship type of thing. Meaning some sort of device that generates an "anti-gravity field," meaning a field that acts like gravity but in the opposite direction. In that kind of anti-gravity two masses (like the Earth and the Sun) would repel, rather than attract, each other. As far as we know, that kind of device does not exist and is impossible to make.

What scientists can do is to make a micro gravity room. That means a place where the effect of gravity is so small that you can pretty much pretend it isn't there. One such micro gravity room is the space shuttle. When the space shuttle is in orbit around the earth the astronauts feel like they are totally weightless. Scientists take advantage of that for doing hundreds of different experiments from the effects of micro gravity on plant growth, to the crystalization of metals, to how humans depend on sight for balance. Another way to get micro gravity is to fly in a plane hat follows a path like the hills on a roller coaster. When the plane is going up it feels like you are very heavy, the plane is pushing against you to take you higher and that feels almost exactly like gravity has gotten stronger. When the plane reaches the top and starts coming down, all of the sudden you start feeling lighter again as the plane is pushing on you less and less. If the plane keeps going into a very steep dive then the plane stops pushing on you entirely and you fall, just at the same speed the plane is falling. From inside the plane then it looks like gravity has dropped to nothing. It only lasts for as long as the plane is falling though, which is about 3 minutes (much longer and it would stop falling and start crashing!). It is pretty hard to do any science in 3 minutes, but NASA uses the plane to get astronauts used to being weightless.

P.S. If you are interested, the idea that the plane pushing up on you feels just like gravity has gotten stronger is called "the equivalence principle," and it is central to Einstein's theories of special and general relativity. If you are interested in learning more about gravity Einstein is a good place to start. He told us most of what we know about how it works.

They don't; it can't be done, according to current theory. The best you can do is fly a large airplane in a parabolic trajectory, so that the contents of the airplane are in free-fall. You can only do this for a minute or two.

Actually, they don't. If anyone could build anti-gravity anything it would be a huge technological leap, one that would change the way people live forever. There are various ways to mimic anti-gravity, though, such as floating in water or riding the free-fall ride at Magic Mountain.