They are inflicted with the incapacitated condition.
Linguistically, this is troublesome, because "becoming incapacitated and unable to stand up for the duration" appears to be a descriptor of "prone", which it is not. "Unable to stand up for the duration" appears to be a more specific ruling of the prone condition (PHB. 292), contradicting the first point:
A prone creature's only movement option is to crawl, unless it stands up and thereby ends the condition.
However, because it is not a descriptor, the only other option is that the save forces the target to fall prone once and inflicts "incapacitated and unable to stand up" for the duration.
It is interesting that this wording would normally prolong the prone condition by removing the option that that creature has to remove the prone condition from themselves, but not by forcing a creature to stay inflicted with the "prone" condition.
This means that while the creature with immunity to the prone condition and targeted by Tasha's hideous laughter does not become prone, it is affected by the rest, and therefore cannot take the action (lowercase a) "stand up," and is also incapacitated (condition).