They pretty much are warnings to potential bidders that the auction item is a step closer to being sold. For example I want to sell my 9 year old child for a starting bid of $5, some guy bids $5 and there are no further bids. I say 'once' which is 1/3 of the way to being sold. I then say 'twice' which is 2/3 of the way there. Now, I can just write 'SOLD' which completes the process but if I decide to wait further for potentially higher bids and someone else bids $10, the entire process restarts and I have to write 'once', 'twice' and eventually, 'SOLD' over again.
TL;DR: They are steps of an auction. Once is the first step to being sold, twice is the second and well, sold is the final.