After you play a game, you should be able to see "Save" in the game lobby after the game is finished. It can be found beside the 'Ready, Kick, Inventory' buttons. After you click save, it will ask you to write a memo so you can remember what the replay is for and whatnot. You don't have to write a memo, it's just there so you can remember what the replay is about whenever you want to watch it.
To watch your saved replays, you click the replay button on the top right beside configuration and then choose a file and click play.