Why SSB output power is limited by peak envelope power, but intelligibility depends on average power. A speech processor compresses or clips the audio so quieter syllables are brought up closer to the peaks, raising the average-to-peak ratio and putting more talk power on the air. The result is a signal that sounds louder and punches through noise better at the receiving end, without exceeding the transmitter's PEP limit.
Watch out The choice about preventing distortion has it backwards: processing deliberately adds some distortion (clipping/compression), and overdoing it makes a signal splattery and hard to copy. Boosting bass actually hurts SSB intelligibility, since communication audio favors the mid and higher voice frequencies.
Processor = more average power under the same PEP roof: louder talk power, not cleaner audio.