Learn Hardcore Japanese: yoru no sora ni takusan no hosi ga miemasu.

空に星が見えます = “You can see stars in the sky.”
By contrast, で marks the place of an action. Since 見える is not an action you perform but a phenomenon that occurs, に is required.

What role does たくさんの play before 星?

たくさん by itself can be an adverb (“a lot”) or a noun (“many”). When followed by の, it becomes an attributive modifier for another noun. So たくさんの星 literally means “stars of many” → “many stars.”

Why is が used before 見えます, instead of は?

が marks the grammatical subject of an intransitive or stative verb like 見える. The stars are the things that “exist in the field of vision,” so they take が.
Using は (星は見えます) would shift the nuance to “As for stars, they are visible,” often implying contrast or topic-focus, e.g. “I can see stars (but not something else).”

What’s the difference between 見えます and 見る?

見る (miru) is a transitive verb meaning “to look at/watch/see” (you actively see something).
見える (mieru) is an intransitive verb meaning “to be visible” or “to come into view.” It describes the appearance or ability to perceive something passively.

Could I say 夜の空は星がたくさん見えます instead? What changes?

Yes, that’s grammatically correct. By making 夜の空 the topic with は, you emphasize “as for the night sky” or set it up as contrastive:
“As for the night sky, you can see lots of stars.”
With に, it’s simply locational: “In the night sky, many stars can be seen.”

Is there any difference between たくさんの星が見えます and 星がたくさん見えます?

Both are natural and mean “you can see a lot of stars.”

たくさんの星が見えます puts the quantity word first, slightly emphasizing “many stars.”
星がたくさん見えます flows more like everyday speech, emphasizing the verb phrase “see many.”
The nuance difference is very subtle—either is fine.