Why does a higher bandwidth allow a higher data rate
Without noise, an arbitrarily high information rate can be transmitted over any narrow-band channel.

In general, the bandwidth depends on the symbol rate, i.e. how often the symbol is changed.If there were no noise, any number of bits could be transmitted with one symbol, since the receiver can reconstruct the signal as accurately as desired. 

In the presence of noise, it is more difficult for the receiver to decide on the correct symbol. The more information (i.e. bits) a symbol contains, the more likely the receiver is to make a wrong decision. 

Actually, any channel has a property, called channel capacity, that determines how many bits per second (or per channel usage) can be reliably transmitted. Claude Shannon proved in 1948 that as long as we stay below this threshold, an arbitrarily small bit error rate can be achieved (at least in principle). 

When we are using FM with a higher deviation (and therefore a higher bandwidth), we actually use a channel with a higher capacity. In the presence of noise, it is easier for the receiver to reconstruct the signal and we will get a better signal-to-noise ratio at the output of the demodulator.