This is an interesting post, because I have seen so many explanations for transistors. Certainly some are innacurate or overly simplified, and others are right on. I guess I can take a shot at it from a different angle, if it helps, and get back to the point of the original post.
First of all, there are more than one kind of transistor(ie FET), but the poster seems to be referencing bipolar transistors, which are IN FACT current controlled. You need to understand how bipolar transistors are physically built, and that will lead you to understand their function. A transistor is BASICALLY two oppositely biased diodes fused together. The point at which they are connected is the base. You should know that a diode consists of two regions, one P-type material and one N-type material, and the point at which they connect is the P-N junction. You really need to study atomic chemistry with respect to silicon and germanium to understand all of this, but the structure of the N region is such that there is an excess of negative ions, and the P region is in excess of "holes". When a diode is forward biased, Being a positive voltage applied to the Anode (P region) and a lower voltage (not necessarily negative) with respect to the anode is applied to the cathode, forward conductance occurs. Reversing the diode polarity will cause only an unmentionable current, which can be considered zero, unless the avalanche voltage is exceeded. The key here is to understand that the voltage value is irrelevant, what is important is the differential between the anode and cathode. You could apply 5000 volts to the anode of a diode, and 4,995 volts to the cathode, and the same amount of current will flow as if you applied 5 and 0. This is why current is more relevant than voltage.
OK, back to the transistor, when two diodes are fused together in oppositely biased directions, you get a P-NN-P junction or a N-PP-N junction. Being the middle or base is fused, we consider it one region, so we call it PNP or NPN. Much like the diode, if a potential is applied to the base(P) of an NPN, and a lesser potential is applied to the emitter (the last N) a current will flow between these regions. Now, I can't properly explain why, but if a supply voltage is applied to the collector that is also more positive than the emitter, and the already mentioned base-emitter current is there, a proportional current will flow from the collector to the emitter. If the base-emitter current is increased, so will the collector-emitter current. THIS IS WHAT IS IMPORTANT TO UNDERSTAND. This is not at all done with voltage in the case of bipolars, it is done by changing current levels, perhaps through a resistance device that is variable..... For instance, a potentiometer can control base current which will proportionally effect collector-emitter current. If the signal applied to the base is of low amplitude, and the supply voltage at the collector is high enough in amplitude, there will be GAIN in current flow, and the tansistor becomes an AMPLIFIER. a small current controls a larger current.
Now, this is an explanation of NPN, in PNP, the opposite rules generally apply. For the base-emitter to conduct, the base has to be smaller in amplitude than the emitter, not necessarily positive or negative, just smaller in potential, and it will conduct, likewise the collector should be at a lesser potential with respect to the emitter, and all the same rules of base current controlling collector-emitter current and gain hold true.
OK, I'm exausted with typing and I think I generally described basic solid state electronics OK. PLEASE, if anyone needs to edit or expand on my comments or correct anything, please do, because I don't mean to mislead. This is just my understanding and I only hope to help someone else to understand. I didn't even go into the difference between common emitter, common base, and common collector, maybe an expert wants to add this. OK, I'm done, fire away!! LOL