Roger beeps aid in the operation on SSB particularly over HF (below 30MHz) radio communications. Due to the fact SSB has no carrier wave, you can't actually tell when the other station has stopped transmitting, however, with a 'roger beep' system the radio can actually generate a beep when the microphone is released. This beep is transmitted after you finish transmitting. This can help greatly during long distance communications when the other stations signal might be poor.
Some modern digitally controlled radio equipment have the option to have an automatic beep when the microphone is released.