The tesla coil is not like most transformers that you may be familiar with. The "standard" transformer uses magnetic fields that are contained in an iron core. These fields transfer the electrical energy from one coil to the other. The voltage gain or loss is governed strictly by the ratio of turns between the coils.

A tesla coil operates on another principle entirely. This principle is called resonance, and is analogous to a fishing pole that you swing with a small motion of your hand. If you move your hand back and forth a small amount at the correct rate (frequency), the tip of the pole will whip wildly back and forth.

Same thing for a tesla coil. If you "swing" it at the right frequency, the voltage at the top will rise and fall wildly.