I wanted to put my answer separate from the main post:
I am not easily invested in a game. About 80% of the games I play are just for fun and relaxation - I often can turn it on and off without a problem, can go weeks without playing it, and may multi task when playing it as I'm not 100% 'in' it. I believe this is because I have read many books, watched many movies and played many games, and am very good at picking up tropes and clichés - when I see an overused one then I get disconnected pretty fast. Other times I just find the motivation lacking, or the character too meh, or the game doesn't feel like it has heart etc. Sometimes everything is perfect, but the controls require so much effort that I don't ever feel like part of the game. For me, pace is especially important - a slow moving game causes my mind to wander while playing, so I never fully connect with what's happening.
Still though, I am affected by some games, and I know what causes it. Surprisingly, it's not story! Some games with powerful moments left me feeling blank. For example, FF7 and Aries (at the time I would have been 12 and easily moved) didn't affect me. Final fantasies are much too slow to keep me completely immersed.
I tend to be most affected when I am taken by surprise. When something unique happens, I fall completely into the game. Unique visuals like that in Okami completely engross me. Music that is more than just background music, but creates a stage and theme, like that in Shovel Knight, keeps me from muting the game or multitasking.
But what matters most to me is how real the main character(s) is. It can be a bunch of pixels on a silent screen for all I care, but if the description/behaviour of the character feels like they were taken from real life, I am completely engrossed. If a main character is a Mary/Gary Sue, silent characters, or 'everyman', I cannot be significantly affected (which is why I will never be fully captured by Mario or Zelda :< ). Even more bizarre, if a game allows you to create your own character like in Skyrim or Mass Effect, I can't become fully invested! I must think I'm boring ._.