This "intrinsic dielectric strength" corresponds to what would be measured using pure materials under ideal laboratory conditions.
If the applied electric field is sufficiently high, free electrons from background radiation may be accelerated to velocities that can liberate additional electrons by collisions with neutral atoms or molecules, in a process known as avalanche breakdown.
Breakdown occurs quite abruptly (typically in nanoseconds), resulting in the formation of an electrically conductive path and a disruptive discharge through the material.
In a solid material, a breakdown event severely degrades, or even destroys, its insulating capability.
In a solid, it usually occurs when the electric field becomes strong enough to pull outer valence electrons away from their atoms, so they become mobile.
The sudden drop in resistance causes a high current to flow through the material, and the sudden extreme Joule heating may cause the material or other parts of the circuit to melt or vaporize explosively.