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Energy
Essentially there are two "bands" in every material which the electrons within the material may occupy - the valence band and the higher-energy conduction band. Electrons in the valence band can be considered as being bound in place, whereas electrons in the conduction band may act as mobile charge carriers.

Conduction band

Valence band
Energy Energygap, Eg

Conduction band

Valence band
 
In dielectrics the two bands are separated by a certain energy gap Eg, corresponding to energies that are forbidden to the electrons.
Energy Energygap, Eg

Conduction band
(vacant)

Valence band
(occupied)
 
Since the valence band is lower in energy, electrons will preferentially occupy this band. Therefore in a dielectric under normal conditions, the conduction band will be empty.
Energy Energygap, Eg e−

Conduction band
(vacant)

Valence band
(occupied)
 
If an electron in the valence band is supplied with energy greater than or equal to Eg, for example from a high-energy photon, it may be promoted to the conduction band.
e− Energy Energygap, Eg

Conduction band
(vacant)

Valence band
(occupied)
 
An electric field of sufficient strength can supply enough energy to promote many electrons to the conduction band at once. Since electrons in the conduction band act as charge carriers, the material now conducts charge rather than storing it.