How rotating generator works: Passing a magnet, or electromagnet, over a conductive material, for example a coil of copper wire, energizes the copper atoms causing lighter weight electrons to come free from heavier weight protons and start jumping from atom to atom and pushing more electrons free in a cascade of electrons going down the wire. Add enough magnetic force, and miles-long wire can be energized. The results is a flow, or current, of electrons available to energize end user loads. Read more

Volts x amps = watts (or power)
Voltage is the force that pushes amperage (or current of electrons) across the wire. Amperage represents the number of electrons.

Voltage generally remains stable throughout the day, but amperage rises and falls, with more electrons flowing when more loads are turned on.
The breaker responds to heat caused by the flow of electrons ... too many electrons on the wire, and it causes the wire to get hot, which trips breaker. It's like having too many cars on the freeway, except with electricity, the breaker trips. So each side of the 30 amp double breaker will allow up to 30 amps. At no time does the breaker carry 60 amps, nor does the load receive 60 amps.
 
In a simplified version of events: As the power plant generator spins around, the electromagnet inside the generator passes over coils of wire, which causes electrons to start moving along the coil. One end of each coil is connected to a transmission line that is energized by the coil. The other ends of the coils are joined to form the Neutral wire that travels with transmission lines. The Neutral represents a midpoint between coils.

The electrons reverse direction every time the north and south poles of the electromagnet passes over a coil, creating the back-and-forth motion of electrons found in AC power (alternating current). The back-n-forth of electrons is called frequency, cycles, Hz or hertz, and represents the number of rotations per second made by electromagnet. In US and Americas, the frequency is 60 Hz, or 60 rotations per second.

The electromagnet passes over each coil at a different moment so electrons on one coil are moving different direction from electrons on another coil.
Since each coil is connected to a different transmission line, each wire carries electrons that are going different directions from each other. So when two wires are connected to a load, the electrons flow back and forth, from one wire, then through the load, going to the other wire, then electrons on both wires reverse direction at same moment and flow back through the load again, at 60 times per second. This is how power is delivered to a load.

A 30 amp double breaker is made for 240 volt. Each Hot wire in the 240 volt circuit originates from a different coil in the generator. When both wires are connected a load that is rated for 30A 240V, then 30 amps worth of electrons will flow back and forth between each wire. So both wires are working together to push and pull 30 amps worth of electrons through the load, and this doubled effort is how the power or wattage is doubled in a 240V circuit, and also shows that amperage going through load is 30 amp, not 60 amp.

If a single breaker is used, then the voltage is 120V, and amount of power going to load is half of a 240V circuit. Why? 120V uses 1 Hot and a Neutral, instead of 2 Hot wires. The 120V Hot wire comes from a coil in generator, but Neutral comes from a midpoint between different generator coils, and therefore carries 120V which is half of 240V. The 120V circuit delivers half power because both wires are not able to push and pull electrons as hard as two Hot wires.