The story
From the sense that movie studios are "dumping" poor-quality and limited-appeal films during those months.
Etymology from Wiktionary, CC BY-SA 4.0.
dump months
Today it means: The two periods of the year, often described as January and February and mid-August through September, when new movie releases are either poor-quality, cheaply made, or intended for niche audiences, since low movie attendance limits the potential box office return. Full definition →