Redeem your product key to GOG.com here. In connection with this, you must either log in to an existing GOG.com account, or create a new one. Please note that all product keys you redeem will be tied to an account and can not be reused so it is important to remember the login information for this account!
After the game is redeemed and added to your account, you will find it under "Account" in the top right corner. Click on your game and choose either to install with the GOG Galaxy client, or manually.
You can use GOG Galaxy by selecting the "Try GOG Galaxy" button, or download it directly here. When you start GOG Galaxy, you'll be prompted to create or log in to a GOG.com account. Once logged in, you will find your games by selecting the "Library" tab in the top bar. Click on a game and select "Install".
If you'd rather install the game manually, click it under "Account" back on the gog.com homepage and download all the files under the heading "Game Downloads." Most games have multiple parts (Part 1, Part 2, Part 3) and all the files are required to install and play the game.