How does the frequency increase and decrease in a FM transmitter circuit?
For FM modulation, it's easier to consider the two functions separately.

First there's the oscillator. Referring to your linked picture, Q1 provides the gain, and L1, VC1 and C2 provide a tapped resonant circuit that gives the transistor the right sort of feedback to oscillate. This oscillation frequency depends on the effective value of each of those components.

Then there's the frequency modulation. The resonant circuit has several extra capacitances connected to it. There's the Q1 collector-emitter capacitance connected to C2, and more importantly, the Q1 collector-base capacitance, connected to L1 via the large fixed caps C1 and C3. These transistor capacitances vary with applied voltage, which includes the audio voltage you're feeding in. 

It's not a big variation, and in professional voltage controlled oscillators we usually use an extra component, a varactor, which has a larger range and a better specified control law, to modulate a VCO. However the effect is always present in transistors, and quite big enough to put audio modulation onto an FM-band oscillator for hobby use as here.