Transistor - Why does amplification happen before the transistor and not after?
The collector of the amplifier  (the pin labeled "C") is actually AFTER the transistor (using the asker's reference frame, which is somewhat misleading.).  The Base (B) is the input, and the active transistor creates a situation where the current on the collector is many times that of the base.  So, the Base current is the input, and the Collector current is the output.

In this case, completing the circuit at the base with your finger creates a small current through the base.  The transistor works things out so as to create a current many times that at the base, lighting the LED.

Why?  I'll punt on the mechanism, but this is what transistors do for a living.  

The figure is taken from http://media.tumblr.com/tumblr_luy74c89IH1qf00w4.png, but probably originates from Horowitz and Hill, The Art of Electronics.  "Transistor Man" looks at the current at the base, and adjust the current at the collector so as to be a multiple of the base current.   Of course, all of this has to do with properties of silicon junctions, but that's a bit beyond the scope of your question.  

The current at the Emitter is the sum of the base and collector currents.

This really is an oversimplification, but it gets at the essence of your question.