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This will bring you to the program window. From here, hit the buy button. Yes, the program is free but you still need to hit the buy button. You won't be asked for a credit-card number, so just click and go.
At the next windows, you're asked to sign into Ubuntu servers using the Ubuntu Single On account. If you don't have one, click on the New account button and get one. It's easy and with it, besides getting access to the Ubuntu Software Center, you'll also be able to use Ubuntu's built-in free, personal cloud storage service, Ubuntu One.
Once logged in, before you can install Steam, or any other program, you must login as the root, or super, user to your local Ubuntu desktop.