The 9v battery attaches by wiring the positive terminal of the battery to the obvious +9V connection point shown and the negative terminal of the battery to the node labelled "ground".
Ground, in a circuit like this is really just another word for "circuit common" which is another term for Battery-. You don't need to connect ground to anything besides the components shown in the schematic. Since this is a high frequency circuit, it is necessary to keep your ground connections short. Of course, as a beginner you don't know how short is short, so use the value of one half inch for now. The higher the frequency, the shorter the ground connections need to be.
As you rightly observed, L1 is an inductor. Like capacitors, we use inductors for temporary storage of field energy, magnetic field for inductors and electrostatic field for capacitors. In this case, the capacitors C4 and C5 are working with the inductor to trade charge storage back and forth on each cycle of the AC at 100 MHz. When you pick the right values of L and C in such a "tank" circuit, the energy storage of the inductor equals the energy storage of the capacitors and the circuit "resonates". When the circuit resonates, you get some interesting things happening, like for example, this particular stage "oscillates" which in fact is what actually generates your FM carrier at 100 Mhz. As you vary the value of C4 or the value of L1 you will vary the resonant frequency and thus will vary the frequency of oscillation, and in turn will move your FM carrier up or down the radio dial.
Q1 and Q2 are the most important components in your transmitter since they are the "amplifiers". These components have the ability to make a small signal bigger. I should not have called Q2 an amplifier because it is confusing to the beginner, but in fact it is an ampifier that is configured to oscillate as mentioned above. So we normally just refer to that stage as an oscillator. There I go using more jargon. You should also learn the word "stage". By this I mean a functional block that is composed of several components. For example, Q1 does not work by itself, it needs support from R2, R3, R4 and R5 to actually work as an amplifier. When I refer to the amplifier Q1 I really mean the stage that includes Q1, R2-R5, and in fact even C1 and C2 which help connect signals from one stage to another. So, to answer your question, Q1 is an audio amplifier that boosts the microphone audio up to a higher AC voltage, and Q2 is your high frequency oscillator that generates the carrier signal at around 100 Mhz. When the audio feeds into the stage composed of Q2, R6, R7, C3, C4, C5 and L1 it will Frequency Modulate the oscillator with your voice audio.