Most chemical drain cleaners contain sulfuric acid or sodium hydroxide (also called lye or caustic soda). These are very corrosive chemicals to metals and organic materials. These chemicals are bases which means they give the clogging substance electrons and the hydroxide ions generate a reaction that clears up the clog. Heat is released by the alkaline chemicals which result in grease turning into a soap-like substance that is easier to dissolve.
Since drain cleaners are a lot heavier than water is, they sink through the standing water that is attached to the clog.
There are other cleaners that contain nitrates, peroxides, and household bleach. Those chemicals cause organic materials that are generating the clog to become oxidized and lose electrons.