Transistor as Oscillator
The cell between the base and emitter is responsible for biasing the transistor on when the switch is open. If this cell was not present, then the collector current would not flow at all when the switch is closed. The voltage coupled through the mutual inductance will be superimposed on top of the DC bias current.

Now, as for why the current increases slowly. The inductor and capacitor in parallel form what is called a resonant tank circuit. This circuit has an impedance that varies with frequency. In this case, the impedance well be lowest at the resonant frequency. The variable capacitor can be adjusted to tune the resonant frequency as desired. Since the impedance is frequency dependent, when you apply a step function by closing the switch, the tank circuit will 'ring' at is resonant frequency. This ringing will decay unless there is something to keep it going - in this case, the transistor.