Why does increased bandwidth give you an increased data rate when sending radio waves? I was told that an increased data rate, with the same modulation scheme, requires an increased bandwidth. I understand that you could send at different frequencies, thereby increasing the data rate. However, I was told that only one frequency was used. Does anyone have a simple explanation for this?
And what happens if you keep the same signal, and just increase the sending rate? Does this automatically increase the bandwidth, or is that something you decide when generating the signal?
 A: A pure, infinitely long, ideal, etc., sinusoid carries no information for it is completely predictable. To be able to carry information it must change in time in some unpredictable manner. That built-in unpredictability is the information. 
It is also true that among many other reasons so that you can easily generate and radiate the EM wave and also have an easily controlled propagation it is convenient to have a signal whose average frequency (roughly its so-called carrier frequency) be much larger than the signal's bandwidth, say $ > \textrm{20x}$. From Fourier analysis we know that for the same type of modulation the occupied bandwidth $B=kR << f_c$ is, of course, directly proportional to the modulation rate $R$ and where the proportionality factor $k$ depends on the type of modulation being used. 
The modulation rate $R=\mu\rho$ is directly proportional to the information rate $\rho$, and the proportionality factor $\mu$ between them (e.g., $\mu \ge 1$ when binary modulation is used) depends on the type of error correction coding used to protect the information from noise induced errors. 
The type of demodulator and decoder needed in the receiver are essentially independent of the modulation rate but naturally will depend on the modulation type and error correction code, resp., being used.
A: Imagine a simple modulation system where you can detect one full wavelength of the signal as one bit. 
If your radio is 100KHz (longwave) then you can obviously only send 1 bit every 1/100,000 second = 100Kbits/s, now if you switched to 5Ghz microwave band you can send a single bit in 1/5,000,000,000 second.
