During the episode "Charlie X", Kirk enters a turbolift wearing a yellow jumper and exits it on to the bridge wearing a green one. I guess he just has a personal wardrobe in there.
I doubt there's a good explanation for the other quick changes in Star Trek... acutally, I was thinking of Doctor McCoy's shiny medical shirt, which seems to turn into his normal, long-sleeved version and back again without any regard to continuity.
Only problem is, when TNG starts, the marvell over how the "Holodecks" and "Holo Technology" are new... So I don't think we can use that as an excuse 74 years earlier...
If my memory serves me correctly, there is a scene missing from the syndicated version of this episode. I seem to remember Kirk stopping by his quarters to change (no, really, I think I read this...). If this is a so, then it is a slip-up caused be the editors who packaged the syndicated versions of the episodes.