All exhaust systems serve two basic purposes: reducing the pollution from engine exhaust, and muffling the sound coming out of the engine. Cherry bomb exhausts are known for their full, booming rumble, but they don't actually create the rumble. Rather, they simply muffle the sound coming from the engine less than normal exhaust systems.
All exhaust systems start with a catalytic converter. The catalytic converter contains a catalyst - a chemical which enables certain chemical reactions. This turns carbon monoxide into less harmful carbon dioxide, and also reduces the output of some other harmful chemicals. It also muffles the sound somewhat by slowing the flow of gas from the engine. The part that provides most of the sound reducing power, however, is the resonating chamber. The resonating chamber has a special shape which reflects sound waves back, cancelling them out and decreasing the volume.
Cherry Bomb Systems
The biggest difference between regular exhaust systems and cherry bombs is that the latter don't have a resonating chamber. Instead, they have a straight metal pipe with holes running down the sides of it. The tube is covered in glass. This provides some muffling, particularly with higher frequency sounds. The low pressure waves from the engine, however, go straight through the tube without much muffling, resulting in a low, throaty growl.
Other Exhaust Tricks
Cherry bombs are often used in a system designed to maximise the flow of gasses and allow as much sound as possible to go through. For example, some of them use partial cut-off systems which allow some of the gasses from the engine to bypass the catalytic converter, increasing the volume. Others use multiple exhaust tubes, which also allows the gas to flow out faster, making more noise.