Assuming I understand your question correctly...
From what I can tell, what gets me to try a game for the first time is a combination of random impulse and what I've heard about the game. The random impulse being a function of my mood at the time, my belief that the game suits my mood, how much the game's packaging (or digital equivalent) gets my attention, and whether or not I think the game will be worth the investment of time and money. If the game has a good reputation (from sources that convinced me I should pay attention to) then that reputation can stimulate my curiosity which has the potential to (though is not guaranteed to) override the other random impulse stuff.
My desire to return to a game after a relatively short period of time (e.g. a day or a week) is usually because I feel like there's something that I want to accomplish within the game. This can be either just one particular mission or level or full completion of the game. It can be either driven by curiosity of what comes next, a desire for a sense of completion, or by simple competition and the desire to do better than either myself or my opponent.
To come back to a game that I haven't played in a long time (say a year or more and assuming no additional updates to the game) is probably most often driven by nostalgia. Competition can also play a factor although this time I may be competing against my past self's accomplishments. Sometimes, I will have just simply forgotten enough about the game that rediscovery of its various elements can be the driving force returning me to play the game. However I think, for me, the nostalgia and rediscovery factors wear off relatively quickly and if I'm going to continue playing after that, I'll probably need some sense of competition to return me to the game (again either against past performance or the opponents in the game).
