* Adverbs modify verbs: The word "brightly" is an adverb, which describes the manner in which the stars shine. Adverbs typically modify verbs.
* "Bright" as an adjective: While "bright" can be used as an adjective (e.g., "a bright star"), it doesn't fit as an adjective describing the verb "shine" in this context.
Therefore, "stars shine brightly" is the more grammatically correct and natural-sounding phrase.