Here is How Does Dry Cleaning Work?

Dry Cleaning is a process for cleaning clothes  using a chemical solvent instead of water. Perchloroethylene or Perc is the most common solvent used.

However Perc is a solvent that is carcinogenic to humans and can cause neurological and respiratory problems from long term exposure and acute also. The Cleaning process in contained in a single machine,  which wets the clothing with solvent, tumbles the clothes around in the same way as a front loading washing machine, then drys the clothes removing the solvent from the clothes and capturing the solvent and recycling itfor later use and disposal.

Clothing does not stay dry during the cleaning process, is just that it doesent get wet with water.

A dry cleaning process also does the following:

Eliminates stains that are not removed during the initial dry cleaning. A process called spotting is used and involves treating stain with various chemicals and using concentrated steam on the stain to remove it.

Press garments using different and specialized steam press.

A Pressing process is an integral part of a dry cleaning service.

Dry cleaning is usually used to clean delicate fibers that cant withstand the rough of washing machines and clothes dryer.

source: Quora

There is also a dry cleaning process for carpets that is used by professional cleaning services and the same principle apply, they used some type of solvent or chemical to clean it.