I'll admit pet battles were the feature for MoP I was the least excited for. Although I didn't talk bad about it, I also didn't look forward to it. I was in college when the whole Pokemon craze hit, and I didn't quite understand how it was addicting, despite having had a friend give me a Jigglypuff toy that sat on my desk for four years.
The very first week of MoP I noticed people getting easy achievements through pet battles, and I thought, "I like nerdpoints, why not round up a few while I wait for this dungeon?" A month and a half later I have a small army of companion pets ready for battle, and their ranks are growing by the day.
What I appreciate about the system the most is its turn-based style that takes me back to the days of Final Fantasy and Dragon Warrior, where I'm plotting what spells to cast and pitting Flying against Aqua. It's not clear right off the bat as you take your favorite level 1 pet around Stormwind or Orgrimmar that there's any sort of strategy involved in the fights; then you find yourself facing off against a couple of spiders in Duskwood and realize you're not going to win unless you use a mechanical pet of similar level. And, of course, you make the mistake of joining a PVP battle and want to cry like your character when all three of your pets are dead at your feet.
Another aspect I enjoy is finding pets in the wild. There are hundreds of them. At this moment, I have alts hovering in strategic positions around Azeroth. My mage is over Jaguero Isle in southeastern Stranglethorn, waiting for rain so I can nab a Baby Ape before anyone else can. I also bring my druid to Felwood when I know a server restart is coming with hopes of caging a Minfernal, one of the most elusive pets in the game, other than the seasonal (only in summer) Qiraji Guardling. Unfortunately, MoP released a week or two after the season, so a Guardling will take a year's wait to capture.
Yesterday, when seeing I had a fifteen minute wait for raid finder, I powerleveled a slew of pets in Pandaria to level 10 for an achievement--I like nerdpoints. I also like taking a low level pet, allowing it to smack a higher level pet, then swapping it out for my ringers and gaining a level each victory. What I'm really striving for is a contingent of pets. I need 3 of each family at level 25, if I'm ever going to clear the Pet Battle Masters in Northrend and Pandaria. I've been stuck in Northrend for a while now, because I don't have the team make-up to beat 3 of the masters and their level 25 pets.
What's even more promising is Blizzard is just getting started with this implementation. In 5.1 they'll be releasing more pets in places such as Naxxramas and giving us rare items to upgrade our captured pets from uncommon to rare, easing the grind of finding those blue titles in the field, which happen to have superior stats.
I think new pets are just the beginning. Down the road, there might even be new families or tricks to capturing pets, such as was done with hunter rares in Pandaria. Pet battles really are an outstanding mini-game, especially when waiting for a dungeon, battleground, raid, or just seeking something new to do outside of dailies. They're not going to improve your character. They're not going to make you rich, because, currently, we're not allowed to sell the ones we capture in the wild (I'd like this to change). If anything, they'll give you some titles, more pets, and nerdpoints to brag about, but after an expansion that took itself too seriously, it's a refreshing change of pace--having fun for fun's sake.