Once you have installed the Steam client, launch the client and enter your login credentials. When you are logged into your account, click the button in the bottom left corner that says "+ ADD A GAME…" to register the product key to your account.
In the pop-up window, select the "Activate a Product on Steam" option. This will open up a new window.

Follow the prompts, and paste or enter the numerical key you copied from your download page into the "Product Code" field.

Hit "NEXT>" and you will be taken to your library where you will be able to download and play your newly registered game!

From your LIBRARY tab, you can see all games registered to your Steam account. You can then select a game and it will bring up a splash page with a large INSTALL button you can click on to install your games. Once your games are installed, the Install button will change to a PLAY button!