Okay let me try to make it simple.
There are three things that are "involved" with the new hextech crafting. 1. Boxes. You get these mystery boxes whenever you or someone in your party gets an -S rank or higher in a game. There is a message after the game stating if someone got an -S rank or higher on your team. You must also own the champion you are playing in order to obtain a chest.
Keys. Key shards are obtained randomly whenever you win a game. This is not guaranteed but it is more likely to get one when playing with a party. Also 3 key shards can be built into key. You must own the champion you are playing to obtain keys.
Champion Mastery. After you play a game, you are given a score based on your performance throughout the game and that is shown in the end game lobby. This will give you point towards a champion mastery on the champion you just played. That also can get you champion mastery 6 & 7 tokens which you can use to further increases your mastery rank.