I don't know that a specific thing got me into gaming. Maybe something did that I don't know of, but I've been playing video games for as long as I can recall. One of my earliest memories is of playing Super Mario Bros 3 in a brightly lit room. My Ocarina of Time playthrough count hit the double digits before I did. If I had to say something, thought, I'd ascribe my fondness of video games to either Halo or Zelda.
For Halo, I loved watching my older brother play, and oftentimes playing with him (read: dying constantly but having fun anyways). I lost interest in the games after a while, though I did take a liking to reading the extended universe novels; but I vowed to beat Halo 3 for the sake of completing the series, as I hate to leave things unfinished. I eventually hit up the multiplayer, and I fell in love. Many an all-nighter was spent on that game, and eventually Call of Duty 4, and from there following CoD and Halo games. If only through the sheer amount of time I've invested into online gaming, Halo truly affected my life.
For Zelda, my family rented Majora's Mask once, and I don't remember exactly why, but we didn't do very well. I myself was too young to really be of any help, and the only ones playing were me and my sisters, who were (well, are) basically useless when it comes to video games. In any case, we only made it to the Deku Palace before we had to return it, meaning that, at the time, that was something unfinished, much like the Halo series a few years after. So, I was kind of subconsciously obsessed with the game. Everything about it fascinated me; the dark undertones, the 4 dimensional aspect of it (the 4th dimension being time, and the 3 days representing said dimension), and of course the desire to know what came after the Deku Palace. I wasn't constantly thinking about it, but I could never forget about it. I eventually got my hands on it again, through the Zelda Collector's Edition on the Gamecube (which is still how I play it); now I firmly hold Majora's Mask to be my favorite Zelda game, and I don't think any future installments could hope to topple its' stronghold. Even to this day, I devote about a week annually in the summer to just sit down and do a 100% playthrough of the game. No shit, there's a special place in my heart for Majora's Mask.