Chhaiyyan is basically another form of Hindi word Chhaon or chhaya. Hence, chhaiyyan chhaiyyan would mean shade shade.
But then we have this system where we put some word twice together and generally it has some meaning more specific than its regular meaning, giving more importance to the word. Here, Chaiyyan chaiyyan, literally shade shade, would mean in shade only. That is, chal chaiyyan chaiyyan would mean walk in shade only (and not in those parts/that time where there is sun).