Sometimes a person avoids greeting first because they want the other person to acknowledge their status. The narration encourages a different habit: offer the prayer of peace without making social pride a barrier.
When two colleagues meet, one initiates السَّلَامُ عَلَيْكُمْ and the other answers graciously. Both behave courteously; the narration specifically praises the initiator.
مَنْ means “whoever.” بَدَأَ is a past-tense verb, and its understood subject relates back to مَنْ.