I have never played the game, but I liked the movie AND understood the plot.

(SPOILER)So Cloud and Tifa have retired from the hero business and are working for the postal service. This is right around the time a disease called Geostigma afflicts children, specifically ones living on the streets. This organization rises up and gains the affected kids' favor by promising a cure and has them join up to fight for them. Also, the silver haired guy is apparently some shell that was used in order to bring back Sephiroth. Cloud and Sephy duel epicly, and then there is a rainstorm that cures the disease.