Leading up to it, I was excited for the release as I loved Zelda prior to it, but the epic scope of the title hit me at three points.
1 - Entering Hyrule Field for the first time. It's small in comparison to open environments in games now, but then it was truly something to behold. Actually having the sun set on you as you traversed the terrain was unbelievable. The sheer size of that field was astounding. To see a hub world (really, that's all it is) presented this way was revolutionary and said "this is a big, epic game you've got here."
2 - Entering Hyrule Market after transforming into an adult. The game really felt over and done with after getting the three spiritual stones. Most games prior would have had that, and nothing more. I was used to the idea that more was to come after the similar structure presented in ALTTP though. I knew more was to come as the titular Ocarina of Time had yet to play a part in the game. The transformation to an adult was nice, but it didn't hit me then as it hit others. What really hit me was coming back outside to see the change Hyrule had gone through. The howling wind, the crows cawing, the redeads screaming and latching themselves onto me... Again, I had seen a transformation of one world into another in ALTTP before, but for some reason this resonated. It felt like something more terrible had happened, and the worst was yet to come. I'll admit I was a bit let down to see this disaster was not as widespread as the Dark World in ALTTP was, but the transformation of the environment after leaving the temple said "this is epic."
3 - The end of the game. During the end credits, I was able to reflect back on what had just happened, and I felt like the whole final boss battle summed it up. It was a pulse pounding, cinematic experience. It really summed up the whole game to me. Combined with the realization that I had used my half-days off in high school to just come home and play OoT until dark, it finally sunk in just how epic the title was, and how much it affected me, and video games in general. It was a game-changer. It was something to experience, not just tell people about.
When people ask what's my favourite Zelda title, and I don't list OoT first, they're confused. The game is good, don't get me wrong, but I've come to love other titles more for what the experience out of time holds (ie - if I played through each game for the first time today). In a specific time period though, the memories of OoT outweigh those of any other for me, and truly make the game epic.