Transformers are used to take one voltage source and create a separate source with a different voltage. This is done by taking two sets of windings about the same core as shown below.
Both sets of windings share the same flux, therefore the voltage across the green wires would be the same as the voltage drop across the red wires, since they have the same F and therefore the same DF/Dt. But the number of red and green loops are different.
Therefore we get the simple expression for the ratio of the voltages.
If the green loop is hooked to a power supply with voltage V1, the red wires can serve as a power supply with an increased voltage if there are more red than green turns around the core.