Why are sidebands generated in AM and FM?
Carrying information requires bandwidth.  

For a given S/N ratio, modulating a signal to carry more information will thus expand its bandwidth.  Call the addition bandwidth "side bands".  If you don't add side bands to a fixed frequency carrier, you can't expand its bandwidth, and thus you can't transmit any information (other than the presence of a constant carrier).

For AM, AM is not PM (phase modulation).  Any additional bandwidth (as required to carry information in the modulating signal) on one side of the carrier will usually have a different phase (change of phase with respect to time from any reference point) from the carrier.  To neutralize this phase difference, AM modulation has to add some additional matching bandwidth on the opposite side of the carrier to carry a signal that will exactly cancel any phase shift of the spectrum the first side, so that AM doesn't become PM.

With FM, modulating a carrier changes the signal frequency to new frequencies.  You can also call those additional new frequencies so generated "side bands".