Karloff's algorithm applied to sparse graphs
I would suggest reading Theorem 5.1 and Lemma 5.1 on the same page in the paper. They answer both your questions.

The proof of Theorem 5.1 in the paper does not use the density of $G$. It will therefore also work for sparse graphs.

You are asking about the motivation behind the assumption of $G$ being dense. In this context the assumption is that a dense graph does not fit into the memory of a single machine. Lemma 5.1 and the conclusions after it show that the input size to the reducers is small enough to fit into the memory.