Why PSK31 shifts phase 180 degrees to send data, and an abrupt phase reversal is a step function that splashes energy far outside the nominal bandwidth. To avoid that, PSK31 shapes each symbol with a smooth sinusoidal (raised-cosine style) amplitude envelope, taking the carrier down through zero as the phase flips. The result is a signal that fits in roughly 31 Hz with very little splatter, at a symbol rate of 31.25 baud.
Watch out Reed-Solomon is a real forward error correction code, but it is used in other systems, not as PSK31's bandwidth control; PSK31's character coding scheme is Varicode, and character coding affects throughput, not occupied bandwidth.
Sharp edges splatter, smooth curves don't: PSK31 rounds its pulses sinusoidally to stay about 31 Hz wide.