I remember going to my friends house when I was younger to play the N64 installments and such, but I never became a so called "fan" until I actually owned a Zelda game.
The first game I owned would be Phantom Hourglass, maybe not everyone's first choice when you talk about a great Zelda game, but I loved it. I have played puzzling games before but very few and even more of a rarity when such games combine it with an engaging adventure -- adventure being one of my favorite things about playing games. So this installment challenged me on a whole different level and it was an experience I loved. It certainly sparked my interest in the series, but was it the sole reason for me becoming a "real" fan? I would have to say no.
Phantom Hourglass certainly made me aware though, and when I bought my Nintendo Wii back in late 2007, I saw Twilight Princess. I would watch videos of time from time to time and my love for that game grew from there. I still remember my first few hours of playing that game -- a moment I will never forget. The music, the visuals, the charm etc. It was all there, here I had a game that had all the elements that I love about gaming.
I won't go on and on about how I love this game, but it certainly made an impact on me and it made me a fan of course. Now I adorned that game more individually than actually loving the entire series as I didn't actually follow the series until the announcement of Skyward Sword, some three years later. However, if it wasn't for this game then I wouldn't be a fan of the series and I feel Twilight Princess similar the perfect embodiment of what a Zelda game is and what most "new" fans should start out with to spark their interest.