Why does Q click (snap) is fingers whenever he uses his powers? Is there an in-universe reason or is it pure theatrics?
I don't know if there was ever an in-universe given by the writers, producers or anyone else but the finger snap was likely that individual Q's method of focusing their power; although as has been seen on mutliple occassions the motion isn't strictly necessary. If you watch all of the various Q episodes you'll see that each Q has their own action they perform when they use their powers (single finger snap, double finger snap, hand wave, etc).
Not ever defined, that I know of, in universe.. but I'd say it's an intentional self-limiting gesture... with near omnipotence and the ability to simply will things into existence, mere thinking could quickly get messy... unless you created some form of action and tied to the concept of 'invoking' your power, at least in your mind... as Xantec points out, they each have some gesture that they use. Imagine the chaos for a new Q until they learn to do this! Every stray thought becoming reality!
This is the same reason the Dresden books have wizards use obscure languages for spells.. once you associate a spell and an effect to a word or phrase, you don't want to trigger it accidentally by thinking it.. so pick something you would not think, except for the specific reason.
I suspect the members of the Q continuum could invoke their powers without their personal gesture... but it might take a pretty strong act of will after a few millennia of habit forming.
Because he is a drama queen, and the rest of the Q collective aren't far behind.
Q only appears in human form only when addressing humans. If Q appears to another race as a member of that race, he does not necessarily have the correct appendages to snap his fingers. Q knows a making things change with a snap of a finger is an impressive show for humans, and milks it for all he can.
It's questionable that the real Q is actually snapping his fingers (or hand waving, or other kinds of fluorishing) - don't forget, we're only seeing a representation of him in this dimensional/existential/plane/thingie.
But, it kind of makes sense - if you were omnipotent, the last thing you want is for stuff to happen as soon as you think it - it would get kind of chaotic (us mere mortals are rather flighty, and I'd imagine an omnipotent being to be more so). So, link it to a physical gesture to enforce the desire for something to happen.
Frankly, I think it's just for show. After all, there are certainly occasions in which Q does not snap his fingers, or make any other noticeable movements. At the end of "Encounter at Farpoint," Q leaves without a gesture, and just after commencing the game of Hide and Seek with Amanda Rogers he does the same thing. Amanda also leaves without apparently making any movements ("True Q").
It may be that young Q are taught to do this in order to first definitively establish whatever it is they want to do, so that every stray thought of theirs doesn't immediately become reality. However, for experienced Q, like most of the Q Continuum, it seems that such movements are totally superfluous. Note that in "Q2," Q vanishes after directing a wink toward Captain Janeway. Q most certainly was trying to be showy in that instance!
In addition to the other excellent answers, I think Q snaps to accentuate the ease at which he uses his powers. The term "it's a snap" refers to something that is very easy to do, like snapping your fingers, so I think this makes sense.