It's surprisingly difficult to find a nice simple description of how a transistor works. This description is from my old physics book - I suspect this may be oversimplified and I'm sure a complete description would run to lots of equations!
Anyhow, this is what an NPN transistor looks like:
so as you say, the collector-base junction is reverse biased and no current flows.
Although it isn't clear from the picture, the base is very thin and lightly doped so the hole density is quite low. As soon as you apply a voltage to the base, electrons flow from the emitter into the base and start combining with holes. These electrons can then cross the base-collector junction and a current flows between the emitter and the collector. As you increase the base voltage further more electrons flow into the base from the emitter, so more flow into the collector and more current flows. This is how the small current between the emitter and the base can control the much larger current between the emitter and the collector.
A PNP transistor works the same way but in reverse.