How can a transformer coupled power amplifier make the peek to peek output voltage twice the supply voltage?

The amplifier is optional, a transformer is an electromagnetic device that can "step" voltages up or down. To achieve this, the input of a transformer must be oscillating. A transformer contains windings around a ferrous material, usually a magnetic steal. There are many shapes, and other specifications for the ferrous material, but I leave this to you to research on your own if you are truly interested. It starts by having two isolated sets of winding wrapped around the same ferrous material. The windings must differ, the ratio between them sets the voltage step. In more efficient realistic cases you need a lot of windings but to make a case example I’ll use small numbers. Lets say you have 5 wraps and 10 wraps (gives a 1:2 ratio). If you were to take the 5 wraps loop and plug it into 120V AC at 60 Hz and placed an AC voltmeter on the 10 wrap loop ends you would see 240V AC at 60 Hz (ideal). Now before you jump to any conclusions there is no such thing as creating energy in such a process, the conservation of power is seen as a give and take between voltage and current. Lets say you were pumping 120V AC at 10A in , you would be able to achieve (in an ideal world) 240V at 5 A out of the secondary windings. The ratio of the secondary over the primary windings multiplies the voltage, but divides the current. Now this coupled amplifier simply means that the amplifier is being used to create the AC from DC and its output is "coupled" attached to a transformer. For other neat voltage increasing devices, look up "step up/ step down " switching power supplies.