I could not find the faulty original schematic at Circuits Today.
1) the value of C4 is so high that it cuts almost all audio frequencies. It should be 0.002uF (2nF), not 0.2uF. An FM transmitter is supposed to boost high audio frequencies, not cut them.
2) Your schematic shows the second transistor upside-down. Its collector is supposed to connect to the LC tuned circuit.
3) Your schematic is missing a very important wire connecting the collector of the second transistor to the trimmer capacitor.
4) There is no way your tangle of wires will work at 100MHz on a breadboard because its capacitance between the rows of contacts is too high and the inductance of your long wires are too high. Use a compact pcb instead.
Here is the fixed original schematic and your schematic: