Why don't you kindly explain to me why I cannot build an RF circuit on a breadboard?
The reason begins with our old friend IxR=V. If you have studied the concept of reactance, you may recall that this is the ability of an inductor or capacitor to resist the flow of AC current. Each inductor or capacitor's ability to resist this AC current can be calculated easily with simple formulae. For example the reactance of a capacitor is stated as:
Xc=1/(2 x pi x f x C)
Where f is the frequency in Hz and C is the capacitance in Farads. Xc is our symbol for the reactance.
Now, if you look at your example transmitter circuit, you will notice that C2 and C5 are very small capacitances. C2 is only 5.6 pF while C5 is variable from 3pF to 18pF. These values are important as they determine the oscillation frequency of the transmitter. What would happen if you put the wrong value in for, say, C2? The transmitter might transmit, but not at the frequency you expect, and so you would not hear it on your FM receiver.
When you use a solderless breadboard, you inadvertantely add a hidden capacitor between every connection. This is because the way the solderless breadboard is built, it has rows of metal strips layed side by side. Because the strips are fairly large and they are laid in parallel, they have a lot of capacitance between them. I measured my solderless breadboards and I find that each row has 3.3pF of capacitance to the next row. Not only that, but from each row to each alternate (second one over) row has 1.7pF and to each second alternate there is 1.2pF. And so on it goes. When there are hidden capacitors in the structure of our board, we refer to them as parasitic capacitance. The solderless breadboard has a lot of parasitic capacitance.
How bad is 3.3pF? Plug that into the formula above and you will see that the reactance is 482 ohms. This means that at a frequency of 100 MHz, the approximate frequency of your transmitter, the most reactance you will ever see from one row to another is 482 ohms, which is quite low. So the RF signal leaks from one row to the other through this 482 ohm capacitor. This leakage from one row to another ruins the transmitter's function. It does this in various ways, and the best way to see how much damage this capacitance does is by adding a capacitor symbol on your schematic between every point where a wire lead plugs into the breadboard. You will see that you are adding many capacitors, in places where they do not belong. Like for example you are adding a capacitor between the collector and the base of Q1. This will really mess up your transmitter's oscillator and cause it to not work. The destruction goes on with every added capacitor.
So, to put it in one short phrase, the solderless breadboard cannot be used for an RF circuit at 100MHz because it adds way too much parasitic capacitance between all of its connections, and these parasitic capacitors ruin the operation of your circuit.
PS: for those interested in this subject, I find that there is also an awful lot of capacitance from each row to the two buses at the top and bottom of the board. For example, from one row to the nearest bus, I measure 1.3 to 1.6 pF. This is even though they are not parallel to each other. From one row to the second nearest bus, there is also 1.3 pF.