The Tesla coil works with the principle of Electromagnetic Induction.

Working of a Tesla coil: Before we start building the Tesla coil, it is very important to know how it works. Only then we will be able to build and debug one successfully. Many alterations were done to the original design to increase the efficiency and usage of the device.
The tesla coil is an air coiled transformer which gives high frequency current and voltage output.

tesla coil working principle pdf