Graham,

[Disclaimer - non-native speaker].

I'm trying to locate a lesson/website where this is explained more comprehesively, but anyway, here's my take on this.

"Vêtements d'hiver" is effectively a noun (in the plural in this case) - "winter clothes".

"Début de l'hiver" is in the possessive ("the start of winter"). As is generally the case in French, we always need to add an article, hence "de l'" in this case (see here: https://www.lawlessfrench.com/grammar/possessive-de/).

If we had said "vêtements de l'hiver", we're in the possessive case, and it would be translated as "the clothes of winter" (poetically I guess this might work!).

Sorry I can't be clearer than this!

I made the same mistake on this exercise but, going through my notes on when to use the definite article with "de", there is a distinction between general things and particular things.  "De" alone connects two nouns where the second noun describes the first noun in a general way like "une table de salon," which means a living room table in general.  By contrast, "de + le/la" describes a particular thing like "la table du salon."  So, I think, "début de l'hiver" in this exercise must mean the start of the specific winter when this story takes place and not the start of winter in general.  In other words, Bastien was waiting for snow since the start of this winter.

Does that make sense?