Semiconductors: Transistor characteristics
...independent of the collector-emitter voltage after some time?

Is the X-axis time? Nope, it is Vce!

In reality the curve is not that flat, Ic increases somewhat with increasing Vce. 

The reason why Ic is somewhat independent of Vce is because the Base-Emitter junction determines the current and the Base-Collector junction has little influence on that as long as the base-collector voltage exceeds a certain value.

That current determined by the Base-Emitter junction is itself set by the current flowing into the base: Ib. Multiply Ib by the current amplification factor beta will give you the collector current Ic. This is only true in the flat part of the curves!

This is not a detailed description of how a Bipolar Transistor works in this respect, I'm ignoring some less important effects.

If you really want to understand the nitty-gritty then you should read a book about semiconductor devices.