A Tesla coil is two LC tank circuits — primary and secondary — magnetically coupled through air. Each rings at its own resonant frequency, f = 1 / (2π√(LC)). Energy transfers from primary to secondary most efficiently when the two frequencies match; the toroid at the top of the secondary sets its self-capacitance, so sliding the tuning control re-tunes f₂ toward or away from the fixed primary f₁.
The coupling k sets how tightly the two coils talk to each other: loose coupling gives a sharp, narrow resonance peak (small mistuning kills the transfer), tight coupling broadens the response but caps the peak gain. Near-perfect matching multiplies the drive voltage by the secondary's quality factor, so a few kV in becomes hundreds of kV at the toroid.
Air breaks down and conducts once the local field exceeds roughly 3 MV/m — so the streamer/arc reaching out from the toroid grows in rough proportion to the output voltage: length ≈ Vout / (3×10⁶ V/m). Turn the drive voltage up, tune the secondary onto resonance, and the arcs visibly lengthen and branch more.