I'm a bit unclear about the structure of the game. At first I thought it was more like Final Fantasy XII's marks, where as long as it is triggered, I can just explore freely and fight any monster I might find on the way. What happens after the hunt fell? I thought I'd return to the base/wherever, but you mentioned carving.
Well it is a game where you gather materials, fight monsters (pretty much bosses, they get harder further along the game, but the game eases you into them), create armor, weapons, and other stuff in helping you complete your quest in completing all quests.
Quests are what you do, it may involve gathering materials, killing certain monsters, and do miscellaneous things (i.e. carry an egg to camp). Quests are initiated in the main hub, which is Moga Village in this game. You choose a quest, and complete it. You are limited in what quests you do based on your "level". So once you complete quests, you will be able to go to the next level, and so on.
So to answer your question of what happens after a monster is killed, when you kill a monster, you get to carve it and get materials for armor, weapons, and jewels (these are basically skill modifiers). You only get a minute to carve, so you have to go fast. Once that minute is over, you are done the quest. You are taken to a reward screen and get rewards, which are basically bonuses for the quest. If you damage certain parts, you will get more rewards.
There is a mode of where you are free to roam around and kill monsters there. Moga village is basically on Deserted Island, which you can roam freely. You can gather materials and kill monsters that may show up.