How do BJT transistors work in a saturated state?
Saturation simply means that an increase in base current results in no (or very little) increase in collector current.

Saturation occurs when both the B-E and C-B junctions are forward biased, it's the low-resistance "On" state of the device. The properties of the transistor in all modes, including saturation, can be predicted from the Ebers-Moll model.