Tesla coils work through the 'magic' of tuned 'tank' circuits - what we call in the trade a "tuned tank (LC) circuit"
LC (Inductor and Capacitor) or 'tank' (resonant circuits) have the ability to store energy, alternately between the magnetic field in the coil and then alternately in the electric field in the capacitor.
Excitation of the secondary 'resonant' tank is done via the un-tuned, low voltage, high current primary; the primary and the secondary are coupled via a magnetic field (magnetic flux).
High voltages are developed via this mechanism: a slow-moving voltage source is applied to the primary, the current builds up in the primary - and then is interrupted, this creates a 'step' current function containing MUCH high frequency energy which is 'captured' by the secondary winding which is linked to the primary magnetically; high voltage is produced according to the well known equation:
e (in volts) = L (in Henrys) * di (the change in current in Amperes) / dt (the change in time in seconds)
di
e = L * ----
dt