Spark Gaps on a Tesla Coil
When the spark gap fires (via dielectric breakdown), the resistance of the spark gap changes from very high to very low.  This effectively couples the tank capacitor to the primary coil, and causes a spike of current.  When the tank capacitor is empty and the primary coil is charged, the collapsing magnetic field will couple energy to the secondary and produce a spike of current.  This spike of current flows through the still-ionised spark gap back into the tank capacitor, repeating the process until enough power has dissipated to allow the arc to collapse.

The ionization of the air is a macro-level event that takes several milliseconds to dissipate, while the primary side resonance is on the order of several microseconds.