Learn Russian: Почему-то стало холодно.

почему means "why". Adding the suffix -то makes it an indefinite adverb meaning "for some reason". You use -то when something happened for a real, specific reason, but you either do not know what it is or choose not to say.

Could I say почему-нибудь here instead of почему-то?

No. The suffix -то is used because this is an established fact—it already got cold for a real reason. The suffix -нибудь is used for hypothetical situations, like in questions or the future tense.

Why is the verb стало in the neuter form? There is no neuter noun here.

This is an impersonal sentence describing a general condition. Since there is no actual subject (like "he" or "she") performing the action, Russian defaults to the neuter singular form in the past tense (стало).

Why do we use холодно instead of the adjective холодный (cold)?

Because there is no noun in this sentence for an adjective to describe. When talking about general conditions, weather, or how someone feels without using a subject, you must use the adverb form ending in -о.