More of a meta-example than anything, but as anyone who went to the midnight releases for X and Y can attest to; for a series that, after the "Pokémania" phase, became something of mockery and ridicule during the first decade of the 2000s, it's both awesome and heartwarming to see so many people show up for games that were a part of their (and many other people's) childhoods. Additionally, it's been said that this game is bringing back Genwunners, as well as many other former fans who stopped playing the game after the Gold/Silver/Crystal generation or thereabouts.
Another meta-example: Mega Charizard X's unexpected reveal, especially for people who believed that Mega Charizard (Y)'s reveal as a Fire/Flying type was the final nail in the coffin for Charizard gaining the Dragon type, considering that Mega Ampharos of all things got to be one. Looks like Game Freak believes in their fans after all!
AZ and Floette's backstory scene. Those graphics were amazing.
The attention to detail is also nothing short of fantastic. For example, when you ride a bicycle, your tote bag will swing on your shoulder as you move. Or if you stand still for a while in Anistar City, your character will yawn and let out a little puff of breath, visible as a mist cloud. Or if you're a girl, and you're wearing a ponytail, it will sway as you walk or ride your bike. The Dev Team Thinks of Everything, indeed!
Taking down Team Flare, especially when you learn the scope of their plans. Their huge, world-wide plans.
Defeating the Flare Boss is a moment of awesome. The guy's ace is a Mega-Gyarados!
Another meta example: Pokémon X and Y selling over 5,000,000 units on their FIRST DAY in stores, setting the record for the highest opening day sales of any game on the Nintendo 3DS, as well as the highest opening day sales of anyPokémon game to date as of October 14, 2013 (by comparison, Pokemon Black And White, which were the previous record holders, had 7,000,000 copies sold over the course of a full week).
If your Pokémon likes you enough, it will refuse to go down from a decisive blow to avoid disappointing you. Some Pokémon even give sort of acknowledgement to your strengthened bond (such as a Charizard briefly nodding its head in your direction) upon entering a battle.
The first time you use Mega Evolution and the lead up is pretty awesome. After battling your rival again and the third gym leader Korrina, you travel to the top of the tower where you meet the latter, who's suddenly acting serious. She then has you use one of her Lucarios on a one-on-one Mega Evolution battle, where she is sporting a new character portrait. After the battle, the Lucario that fought with you refuses to leave and she allows it to stay at your side with the Megastone.
Expanding on the lead up, learning about Mega Evolution was the mission Professor Sycamore gave you at the start. You've been seeing Korrina off and on throughout the early game, with her two Lucario. And Lucario's liking for you was well established in his first appearance.
The fact that the finale averts Dude, Where's My Respect? with gusto is this. Usually, whenever the Player Character saves the region's ass from the Villainous Team, it is quickly forgotten. Not in Kalos. You and the other four friends get the Pokemon equivalent of the medal of the Légion d'Honneur and are treated to a huge parade of people to celebrate their new Champion as well as the fact the five took down Team Flare. Bravo!
Also a Crowning Moment of Heartwarming, even without AZ's storyline. It was all Professor Sycamore's idea to congratulate you, and the parade is absolutely packed. It feels really nice to be that appreciated for a change.
Even better, some venues in Lumiose City allow you access specifically because you're the champion. Sushi High Roller, here I come!
If you have a high level of affection, you will occasionally get messages from your Pokémon at the start of battle. For certain plot-related or important battles, like gym matches or the Elite Four, your Pokémon will acknowledge the importance in the most epic way possible:
Also going with having maxed out 5-heart affection: sometimes, your Pokemon will even 'read your mind' before you call out the attack, thus avoiding those pesky first-hit moves like Fake Out or even moves that would have KO'd it!
When you enter the Poké Ball Factory, a Team Flare grunt will be about to challenge you, but Serena/Calem gets a minor one when they come in and asks the grunt if she had to work at the factory to pay for her "ugly suit"; also a Crowning Moment of Funny.
Honedge/Doublade/Aegislash can use Shadow Sneak, a move that uses one's shadow to attack. Essentially this could be taken to mean that the attack involves stabbing at your foe with a sword of shadows. That just sounds awesome!
Any time your Pokémon Mega Evolves, especially if it's one of your personal favorites. The cutscene is awesome, and so is your new powered-up monster pal.