If FM radios work by modulating the frequency, how is it that we can tune into a specific channel, and hear a song or station? Wouldn't the channel need to be modulated along with the varying frequency?
The frequency of the signal is modulated in a relatively narrow band, and drives the audio circuit in proportion to the resonant power between the signal and a resonating circuit tuned to almost the range where the signal resides.
The result is that the power in the audio circuit varies with the frequency of the driving signal
Other signals are far from the resonant frequency of the tuner and do not drive it to an appreciable degree. Further regulatory steps are taken in most place to reduce insure a minimum difference between nearby stations.
Thus, in the design of any radio demodulator circuit, the bandwidth of the modulated carrier must be taken into account.
Generally, the RF signal is down-converted, via a mixer and local oscillator, to an Intermediate Frequency (or more modernly, the RF is directly converted to baseband) where a sequence of filters and amplifiers, with the appropriate bandwidth, attenuate frequencies outside of the desired bandwidth and amplify the frequencies within. This "groomed" IF signal is then passed to the demodulator.
For the commercial FM band, the channel spacing is 200kHz. But, the actual bandwidth is (from Wiki):
Bandwidth of 200 kHz is not needed to accommodate an audio signal — 20 kHz to 30 kHz is all that is necessary for a narrowband FM signal. The 200 kHz bandwidth allowed room for ±75 kHz signal deviation from the assigned frequency, plus guard bands to reduce or eliminate adjacent channel interference. The larger bandwidth allows for broadcasting a 15 kHz bandwidth audio signal plus a 38 kHz stereo "subcarrier"—a piggyback signal that rides on the main signal. Additional unused capacity is used by some broadcasters to transmit utility functions such as background music for public areas, GPS auxiliary signals, or financial market data.