Let me rephrase your questions:
1)
In the saturation region, why does Ic increase with an increase in Vce?
2)
In the active region, does the graph indicate that a specific base current only provides a specific collector current (that is, Ic=Ib * Beta)?
Answers:
1)
This is one example of a real device versus an ideal one. Examine the graph and you'll see that Ic increases very rapidly with a small change in Vce in the saturation region. In an ideal device, any Vce greater than zero would suffice, but in a real device the physics of the junctions require some Vce. In practice you typically don't want the Vce to swing too close to the Vce(sat) specification.
2)
Basically correct. Once the transistor is in the active region the collector current (Ic) becomes (roughly) proportional to the base current (Ib). The amount of current flowing into the collector vs that into the base (Ic/Ib) is roughly the transistor beta which is specific to the device and doesn't radically change for a certain specified range of operating parameters which includes Vce (although it will change depending on certain conditions). As such, you can estimate collector current via Ic=Ib * Beta in a test setup (note that beta is rarely used in this specific fashion for actual design purposes).