To emerge out of thin air means to:

Appear suddenly is the correct answer. The idiom "out of thin air" means to appear unexpectedly or seemingly from nowhere, without any obvious source or warning.

• Fall down quickly is incorrect because the idiom refers to appearing, not falling.
• Descend gradually is incorrect because it means to move downward slowly, which is unrelated to the idiom.
• Appear suddenly is correct because it accurately expresses the meaning of the idiom.
• Enter from air is incorrect because the idiom is figurative, not literal.