As missions are quite new, I can only tell from my experience:
How missions work in general
Every mission explains itself in its short description. Most of them are "win a complete match on this specific map" or "get a number of kills in this specific game mode". After you completed this mission, you get rewarded with a weapon drop.
A little bit more in detail
You do have 2 missions at start. Every mission is explained quite simple and in order to achieve it / get your reward you need to finish the map you would complete the mission - so don't leave to early.
For example: "Get 6 kills in game mode Destroy." This means you have to look for a game in this game mode. You will join a server and you likely will kill 6 people. After you have killed those guys, be sure you stay until this map is finished.
In the end screen score board, you will receive your mission reward as a weapon drop. (Since now I only got trash drops, I don't know if they cover up uncommon or rare items, too. All items seem to be out of the new added items, no old ones.)
How to get new/more missions
For this you only have to play one supported game in the finding a game screen. This covers up match making, Gungame, Destroy and so on. In the end of the game, where weapons drop, you will now have an additional chance of a mission drop.
So if you need some missions, you may play a few Gungames, as they're finished quite fast and you get the chance for a mission drop after each game.
Best Answer
This is based on how many competitive matches you win, and hours spent in both Competitive and Casual servers (Official Valve servers only!)
From wiki.