How do we make use of the saturation region? How would it be applied in a problem?

Well, most problems we'll encounter will have the transistor in ``normal operating mode'' with the $be$ junction forward biased, and conducting current such that the Transistor Man equation, $I_c = h_{fe} I_b$ applies (this is the plateau or ``linear active region'' of the $I_c$ vs $V_{ce}$ plot).

The saturation region is most relevant for the switch; it refers to the case when both pn junctions are forward biased and a large current, approximately independent of $I_b$, flows from collector to emitter. See also above question.