According to the paper, the n-gram model predicts the next word based on the previous n words. Let's get intuitive...
Say n is 4, and you have "I love to eat". That is, given "I love to eat ______", you want to fill in the blank with the most likely word.
Intuitively, I would guess that "I love to eat ____" and "I like to eat ____" would have a similar answer and thus "love" and "like" might have a similar meaning in this context. As far as I can understand it, the algorithm is trying to determine that {"love", "like"} are a word cluster.
By definition, you'll have more examples in your corpus of {"love", "like"} than you will of either "love" or "like" individually, which means your data will be less sparse when you use the cluster rather than individual words, which also helps with calculations.