Do the names of the seasons start with a capital letter?

It's not a great explanation but the simple reason is "because you don't." It's one of the quirks of English as it evolved. Admittedly the seasons are no less proper nouns than--for example--the days of the week or the months in the year, however it's customary to not capitalize the initial letters.

Historically "winter" and "summer" were units of measure and so they weren't capitalized. "Spring" and "fall" were part of phrases describing the periods, and since we don't capitalize verbs those, in turn, remained void of the initial capitalization. As for "autumn," it's just an odd one.

However since we've well moved past dealing with them as units of times or parts of a phrase we come back to the regretful answer of "because you don't." English has many quirks, and this is one of them.