Learn Russian: Это птичье гнездо.

It is a possessive adjective formed from the noun птица (bird). The dictionary form is птичий. Because it modifies the neuter noun гнездо (nest), it must take the neuter ending -ье. Notice the soft sign (ь); possessive adjectives in this group use it in all forms except the masculine nominative and inanimate accusative.

Can I just say гнездо птицы instead of using a possessive adjective?

Yes, гнездо птицы (nest of a bird) is perfectly correct. However, Russian heavily prefers these -ий/-ья/-ье possessive adjectives for things naturally belonging to or produced by animals, like habitats (птичье гнездо), body parts (медвежья лапа), or food (коровье молоко).

Does the soft sign (ь) in птичье actually change the pronunciation?

Yes, it acts as a separating soft sign. It tells you to keep the soft ч sound distinct and then pronounce the е fully as 'ye' (like in 'yellow'). If the soft sign were missing, the two sounds would blend together into a simple 'che' sound.

Since гнездо is neuter, is это taking a neuter ending here to match it?