How do I send the winner their key?

the best way to do it imo is to reveal the key yourself and enter the key when you're making the giveaway and then when the giveaway is over you just click the link in the sent column https://www.steamgifts.com/giveaways/created and then it takes you to the giveaway winner section and then you send the key

Just check the box to send it. It's like a final confirmation just in case.

There's a reason the key/link is not automatically sent. On several situations, the giveaway maker wants to check stuff. If the winner passed the SGTools gate, if they own the game but didn't sync SG with Steam (happens), if it's a bundle then what amount of it the winner has, and more. Thus, for every giveaway you make, you have to manually approve and send the key/link.