Chain cleaners are designed to break down the build-up of grease and dirt that gathers within the drive-train, between chain links, the cassette and derailleurs. It is important not to let the drive-train gather too much muck as this will start to affect the efficiency of the bike. When gunk builds up on the chain, it may start to slip and could cause both the front and rear derailleur to become blocked with dirt and may seize - which could be dangerous when riding.
Keeping the drive-train as clean as possible enables you to keep a close eye on your bike, helping prevent any mechanical surprises whilst out riding.
To assist with drive-train cleaning a useful addition to your cleaning kit would be a claw brush designed for the cassette, making it easy to clean down to the hub and push dirt out. A chain cleaning tool (bath) is also very useful and there are also chain sponges with a cut-out slot for the cleaning of chains - when the chain bath isn't required for a complete clean. Owning one or both of these will give your drive-train multiple leases of life.