I didn't understand it either, but now given the sibling posts I'll give it another shot:
Assume you have a number of hot dog vendors in a stadium, and over time randomly people get hungry and want a hot dog. One of the hot dog vendors needs to come to them, covering a certain distance, and "serve" them their hotdog. (After that, the vendor will idle around there.)
Assume there is a central controller with a radio that oversees the whole thing, noticing requests, then picks a hot dog vendor when a request comes in and sends them on the way. After the game, all the hot dog vendors together walked a certain distance: that's the cost (which of course you'd like to minimise).
Crucial question now is which hot dog vendor to pick for each request, and there are many algorithms (you could always pick the closest one, for example).
However, now comes the trick: Suppose the controller knows in advance all the requests - who will want to have a hotdog when and where. He still, anytime a request comes in, needs to pick a vendor to send them to the request. But now, knowing the entire future, the controller can make better choices, leading to a smaller total cost. (That's the offline version; getting to know the requests only "as they come in" is the online version.)
The question now is: Compare the actual cost an "online" algorithm incurs with the "super optimal" that would have been feasible with full foresight ("offline"). It was proven that, for k hotdog vendors, it is at least k times higher (that's the "competitive ratio") worst case (plus a constant). On average, the online algo can do much better, but worst case it would be at least k times worse.
Here, the authors of the paper prove the conjecture, namely that it is also at most k times higher. (So, even if an evil genius plans the sequence of requests against this algo, it can't make it more than k times worse.)