Ozone sounds plausible. I've been getting a variety of answers from other search engines, ranging from "fabric softeners! Use our products!" to "depends on whether or not you live near a farm....". One mentioned geosmin, a natural essence in soil, and commonly tainting drinking water taken from surface supplies.
In your answer, is the ozone formed by UV action directly at the surface of the drying clothes? Why does the fragrance persist for several days on the clothes rather than just blowing away?
Either way, it's one of the great rewards of outdoor life!