The creature falls down before becoming prone.

A flying creature cannot be rendered prone. However, the FAQ on flight states:

Flight and Magical Flight: Can a paralyzed or stunned creature keep flying with magical flight? Does a creature with magical flight not apply bonuses or penalties to Fly checks because it doesn’t have a “natural” fly speed? Does flying make a creature immune to being flat-footed?

No, any creature that loses all actions can’t take an action to attempt a Fly check to hover in place and thus automatically falls. That includes a paralyzed, stunned, or dazed creature. Magical flight doesn’t act any differently, even for paralysis, as it isn’t a purely mental action. A creature with 0 Dexterity can’t fly, and paralysis sets a creature’s Dexterity to 0. Despite the fact that the Fly skill mentions that bonuses and penalties from maneuverability apply to creatures with natural fly speeds, they apply for any fly speed. If they didn’t apply to creatures that gained flight artificially or through magic, then those maneuverabilities (like the listed good maneuverability for the fly spell) would have no game effect. Finally, the statement “You are not considered flat-footed while flying” means that flying (unlike balancing using Acrobatics or climbing) doesn’t automatically make you flat-footed or force you to lose your Dexterity bonus to AC; it doesn’t mean that flying makes you immune to being caught flat-footed.

Since the creature cannot take any actions it thus cannot hover and falls down as per the normal falling damage rules. This means that if they hit the ground after falling 10 or more feet, they take the appropriate falling damage and are rendered prone automatically. If they fall less than that, they are rendered prone from the spell.

Incapacitated will prevent opportunity attacks

Your player is correct. The opportunity attack requires a Reaction. And Tasha's Hideous Laughter's description states (PHB, p. 280, bold added):

The target must succeed on a Wisdom saving throw or fall prone, becoming incapacitated and unable to stand up for the duration.

And the "incapacitated" status states (PHB, p. 290)

An incapacitated creature can’t take actions or reactions.

So if someone is under the effect of Tasha's Hideous Laughter, they can't take the Reaction necessary to make an opportunity attack. They'd be able to crawl and do things that don't require actions or reactions (or bonus actions) such as speak, but can't do much else. They'll get a chance to break this spell with a Wisdom Saving Throw at the end of their turn, and any time they take damage.

NOTE: The "prone" condition doesn't prevent a character from making opportunity attacks (they will just make them with disadvantage). It's the "incapacitated" part that stops them.