ELI5: Why does Fairy typing give Azumarill a chance at OU?
Also: What are terrain moves? What is terrain?
Fairy types have good resistances and good type coverage. Being immune to Dragon and resisting Fighting, arguably two of the biggest offensive typings, is a huge defensive boon. In addition, Fairy type STAB can hit the same Dragon, Fighting, and Dark types, all fairly common and powerful typings, super effectively. Azumarill was already pretty good, but the Fairy typing might be the last thing it needs to get a usage boost.
Terrain moves are akin to Trick Room in that they give added effects for 5 turns. Their catch is that they only affect grounded Pokemon, so Pokemon that hover are not affected. They give a variety of effects, such as Misty Terrain halving Dragon type move damages for grounded Pokemon. This isn't its only effect, and there are terrains. They can stack with standard weather.
After you beat the elite four, where is a good place to level up fresh pokemon to 100? I know the looker scenario only last so long, and i heard about cafes being good permanent battle spots, but i have no ideahow they work or how to pick a cafe in the right level range. Any help and explanation would be appreciated. Also, i am horrible at chaining pokemon.
Your daily rival battle is a pretty good experience location. The starter hits level 70. Otherwise, grinding the Elite Four, although tedious, is probably your quickest bet. Use a Lucky Egg to speed things up just a little (and maybe exp. boost O-powers).
To answer your question though, there are 1, 2, and 3 star restaurants in Lumiose, along with a fancy sushi place. They each cost money to participate in. For example, the 3-star restaurant requires 100,000 and Sushi High Roller requires 500,000. I can tell you the sushi is not worth it, haha. In the 3-star Restaurant Le Wow, you pay 100,000 to start a "meal", which is just a sequence of 5 people coming in the battle you. They have Pokemon around level 63, and you can choose Double, Triple, or Rotation battles. You win Balm Mushrooms at this restaurant when you finish the course, and you can sell them for a good profit (more than it cost to get into the restaurant). The 1-star and 2-star restaurants Le Nah and Le Yeah only have Pokemon around level 10 and 30. Sushi high roller might have slightly higher levels, but they do not break 70 and the rewards do not make up for the steep entrance fee.