Note: a previous version of this answer held an opposite opinion based on incorrect information, which I no longer agree with.
Fairy Flight is suppressed
The first two paragraphs of Antimagic Field indicate that it is likely intended to affect all magical effects except the absolute most powerful kinds (artifacts and divine intervention basically). More importantly in this case, it also provides a list of very specific ways it affects very specific types of magical effects. This case falls under the "Spells" heading:
Any active spell or other magical effect on a creature or an object in the sphere is suppressed while the creature or object is in it.
Even though Fairy Flight is not a spell, and thus someone (like myself) may incorrectly assume the information under the "Spells" heading is not applicable, the actual text says "Any [...] magical effect on a creature [...] is suppressed". Fairy Flight is a magical effect as you noted, and it is definitely on a creature, thus it is suppressed within an Antimagic Field and the Fairy will be unable to fly.