Understanding a simple crystal oscillator transmitter circuit
I post this very useful answer which comes from a fellow ham who still believes that helping people to understand is part of the noble art of amateur radio. Someone who posted comments above should think about it.

Start by thinking of this circuit as an amplifier.  The 27 ohm resistor from the emitter to ground (negative terminal) puts a limit on how much current will flow.

The 10K resistor from the base to the positive terminal puts a positive voltage on the base and biases it so that current will flow through the transistor.

Now the fun begins!   It is an amplifier, but it has no input signal!  The input signal is the output signal -- it is like a dog chasing his tail!

The crystal is very important.  It is the main frequency determining element, and it is the conduit for the feedback that gets this thing oscillating.  It is a piece of quartz.  If you put a voltage across it, it will begin vibrating (physically) at a specific frequency.  As it physically vibrates, it also creates electrical vibrations.

So, when you turn this thing on, noise in the circuit will put a bit of charge on the crystal.  It will begin to ring, much like a musical tuning fork.  The electrical vibrations from the crystal will go to the base.  They will be amplified by the transistor and will emerge (stronger) from the collector.   From the collector, they go to the  tuned circuit formed by the big coil and the variable capacitor.

The coil wound on the film box serves several purposes. The portion of the coil between the positive terminal and the collector carries the 12V DC to the collector of the transistor.  It also carries the amplified RF signal coming from the collector.  This signal goes through the lower portion of the coil and causes the coil and the capacitor to resonate.  The signal at the top of the tuned circuit peaks when the tuned circuit is tuned to the resonating frequency.

The capacitor/coil tuned circuit (with the tapped coil) are set up so that the right amount of energy is fed back from the output to the input, and that this energy is fed back in the proper phase relationship to the signal at the input.  Think of a child's swing at a park:  To keep the swing oscillating, you have to push at the right moment (frequency and phase) and with the right amount of energy. 

The little capacitor across the battery is to prevent "key clicks."   The output coil on the main coil takes some of the energy and sends it to the antenna while converting the impedance of the antenna to a suitable "load" for the transistor.