E2 occurs in a single step. Halide, transition state, alkene...
If we were to set up an experiment where the base (let's say alkoxide and conjugate acid) and solvent (simply the alcohol) were the same and the only thing that differed was the the halide, we see that the tertiary halide reacts the fastest.
The potential energy of an E2 transition state is closer in potential energy to an alkene than the starting halide. According to Hammond's postulate, the E2 transition state "resembles" an alkene to a greater extent.
Using the above as a guideline, E2 transition state stability should correlate with alkene stability (again, by way of hammond's postulate).
Tertiary halides have the greatest potential to generate alkenes with the greatest alkyl substituents and hence the most stable alkenes. Therefore, tertiary halides will undergo E2 the fastest because they yield the most stable transition states (lowest activation energy).
This also explains why in MOST cases Saytseff elimination is seen. The more substituted alkene is PRECEEDED by a more stable transition state and thus forms FASTEST.