How does linear phase shift of bessel filter avoid distortion of nonsinusoidal inputs?
How does linear phase shift help here? I'm familiar with some fourier
  series but I don't get my textbook explanation. Any help?


A linear phase shift (as produced by a Bessel filter) means a constant time delay for all the main frequency components of a signal. So, for instance, if the filter were a 2nd order low-pass type having a 3.01 dB cut-off point at (say) 12.64 kHz: -


159.2 Hz would see a phase shift of about 0.99 degrees
1.592 kHz would see a phase shift of about 9.86 degrees
7.977 kHz would see a phase shift of about 49.02 degrees


Then, if you factored in the time shift you would get: -


159.2 Hz has a time shift of 17.27 us
1.592 kHz has a time shift of 17.20 us
7.977 kHz has a time shift of 17.07 us


In other words, a 2nd order Bessel filter is good for constant delay up to around 50% of the cut-off frequency AND, that will largely maintain good shape with maybe about 0.5% overshoot in the time domain from a step change.

You can use this on-line tool to check out the numbers. Manuipulate the R value to give a \$\zeta\$ of 0.86. I used R = 172 ohms knowing that a 2nd order Bessel filter has a damping ratio of about 0.86.