The quick answer is that a liquid lawn aerator works chemically to break the bonds of clay soil. This reduces the soil compaction that plagues owners of a clay lawn or garden. The more detailed answer is that chemicals (humic acid, polymers, and such) work to make the soil less compact while encouraging the worm population to work hard. These aerators do work, but you need to start with reasonably loose soil to begin with. They also help to break up sodium which can contribute to soil compaction.
This stuff comes concentrated, and in one of those bottle things that you attach to the hose. Just start spraying and you soil will be nice and soft in no time. You should use it a few times per year so things don't get out of control.
Although this will help with the compacted soil it will do nothing for thatch. If you normally aerate to help control thatch you'll need to either manually dethatch your lawn or just get a dethatching rake to use in the spring or fall.
If your soil is badly compacted and hasn't been aerated in awhile do it the old fashioned way first and then follow up with the liquid aerators. That way you'll get a nice penetration of the soil and the liquid will go deeper.
Aerating your lawn will help water and other nutrients to go deeply into the soil. This encourages deep rooting grass and will make it more drought tolerant. You won't have to water as often or as long which will save on your water bill and help conserve the ground water.
When you do water, make sure that you water deeply. The aeration allows the grass roots to wander through the soil but watering deeply gives the grass a reason to go there. Shallow watering encourages shallow roots which is a bad system.
If you've got serious clay soil grab you a lawn aerator and some liquid lawn aerator and just have a nice party in your yard. Make it a part of your lawn care schedule and you'll have a happy heathy lawn.