Use of Big O Notation in a recent paper by Khot et al
In general, when you see big-O notation in algorithms, the default assumption is that the number that's varying is the size of the input.  Here the input size is $n+m$.

In this case the claim is valid if you think of it in terms of $m$, the number of constraints.

The claim that is being made here is a very basic claim.  It's basically a consequence of the fact that $\text{Var}[X] \to 0$ as $m \to \infty$, where the random variable $X$ is the fraction of constraints satisfied by a random assignment.  (Heuristically, if you were willing to assume independence, it would follow from the Central Limit Theorem.  In this case, you cannot assume independence, but a similar result follows nonetheless.)

If you are not sure how to prove this claim, or were unable to recognize this as a fairly standard concentration result, then you're probably going to find the  rest of the paper very difficult going, and you might  need to start by reading more basic textbooks and papers before diving into this paper.