US domestic supply systems are split-phase. So homes are supplied with 2 phases or hots and a neutral. Each hot is at a potential of 120 volts with respect to ground. Lower powered appliances are fed from 120 volts, and higher powered appliances are powered by 240 volts which is derived from the voltage between the two hots.
This is a 2 pole common-trip breaker used to supply 240 volt appliances. The inputs to the breaker are fed from 2 different phases or hots. Each of the hots is at a potential of 120 volts with respect to neutral. Either of the output terminals of the breaker can supply 30 amps max to a circuit at 120 volts. That's 30 + 30 = 60 amps total to two separate 120 volts circuits. However this type of breaker is usually used to supply 240 volts circuits (There is 240 volts between the two output terminals) and will supply 30 max to a circuit. The reason the operating handles are joined is so that if one phase sources too much current, the breaker will trip and shut the other breaker off also. This ensures the circuit is totally isolated from both hots. A dangerous scenario can occur if a 240 volt circuit is wired from 2 single breakers. If one of the breakers trips due to an overload (and the other breaker doesn't trip) or if someone turns off a breaker to do maintenance, 120 volts can still be present, presenting a shock hazard.
I hope this is accurate! Maybe our resident electrician, Wilderness, can add to this?