Tesla coils are roughly divided into three parts. They're Primary resonance circuit, Secondary coil, and Power supply circuit.
First, a direct current is generated in the power supply circuit, and the capacitor is gradually charged through the resistor. While the capacitor is charging, the voltage at both ends rises. At this time, the same voltage is applied to the discharge gap.
When the voltage of the discharge gap reaches the spark discharge voltage, pulsive arc discharge occurs, the capacitor and the primary coil are connected via the discharge gap, and an oscillating current is generated in the capacitor and the primary coil.
An oscillating current flowing in the primary coil creates a magnetic field or a spiral electric field around the induction, and induces the secondary coil. This induction does not cover the entire secondary coil, but it strongly guides the portion close to the primary coil.

