Lets be clear here - we are only talking about bipolar junction transistors in this thread - there are other kinds too(!).
I think the best intuition about bipolar transistors is that the collector current is a large multiple of base current (unless saturated) and the Vbe is about 0.5V. That's all you need to design most circuits really.
If you want something a bit deeper then think of the base as a tax-collector on the collector current - it takes its percentage or nothing is allowed to happen.
Of course the better explanation is about minority charge carriers flowing across the base, but that explanation only helps you design a better transistor, not use one.