A telephone headset performs the functions of a telephone handset, but it's worn on your head rather than held in your hand. It's much more convenient and comfortable to use a headset, than to hold a handset or to squeeze it between your cheek and your shoulder to free your hands. It's not very efficient to type with just one hand.
Most modern headsets are comfortable for people who wear eyeglasses.
The amplifier is the electronic interface between the telephone and the headset. Years ago, all telephone calls were routed through manual telephone exchanges. Calls were connected by an operator who used plug-in cables to physically patch lines together at a switchboard. Headsets made life easier for the operators, by enabling them to use both hands to to insert and remove plugs, and write down information about phone calls on billing tickets. The handsfree format was very convenient, and headsets became common in TV studios, airplane cockpits, submarines, space ships, race cars, fast food restaurants, football stadiums, and everywhere else.