Welcome to the site and thanks for sharing! :)
After your giveaway ends the site will automatically determine a winner. You will have a banner on the site telling you that one of your giveaways has ended. Clicking it will take you to a page that shows you the winner and his e-mail address. At this point you have the option of e-mailing the winner (be it a steam gift or a key) or adding them to your friend list on steam and handling it there. Whichever way you choose, you have a week to contact the winner. After receiving the gift, the winner will mark the gift as 'received' on the site and you will gain contributor value for the giveaway. I'll leave out the details on contributor value and bundled games.
The biggest rookie mistake is probably people that make a giveaway thinking that the game will magically appear and be gifted to them. Making a giveaway means you are giving a game away and obviously before doing so you should have a giftable steam copy or working key to give away, first. :)