First you have to understand why impedance matching is required for maximum power transfer. Consider the circuit below:-
simulate this circuit – Schematic created using CircuitLab
The source has an output impedance of 50Ω and the load is also 50Ω, so voltage splits 50/50 between source and load. As a result only half the power is transferred - the other half is lost in the source.
This is only 50% efficient, but it is also the maximum power that the source can deliver to the load. Why? If the load impedance is increased then the voltage transferred is higher but current is lower, so the power transfer is less (down to zero Watts at open circuit). If the load impedance is decreased then current is higher but voltage is lower, so power transfer is also less (down to zero Watts with a short circuit).
The purpose of a matching transformer is make both the source and load 'see' the impedance required for maximum power transfer. So if the load is 200Ω then it wants the source to also be 200Ω, while the source wants the load to be 50Ω. A 1:2 transformer achieves this by doubling the input voltage, which is exactly what the 200Ω load needs to draw the same power as a 50Ω load at half the voltage.