August 13th through 16th are called Obon holidays. Obon is traditionally a Japanese Buddhist custom to honor the spirits of ancestors. Like Thanksgiving holidays in the United States, the historical meaning is being lost and many people simply consider it "summer vacation".
So what's Obon got to do with Mount Fuji? Not only the mountain, but also every traveling-related spot in Japan, from sight-seeing places to train stations, will be very, very crowded. If you are living in Japan and working for a Japanese company, you probably don't have any choice. This is one of the few occasion in a year when we can get (relatively) long vacation. If you are traveling from foreign countries, Obon may not be the best time to visit Japan unless you want to experience the most crowded condition of Japan.