How does giveaways works here now?

Lots oof other sites to buy steam keys from, other than Steam.

Fanatical
IndieGala
HumbleBundle

Just check out isthereanydeal.com to see where to get the cheapest Steam keys at any given time.

Well, you can just add the winner of a giveaways as a friend and gift them the game.
But as far as I'm aware, you were never able to buy keys directly from Steam? You could buy a steam gift to your inventory but that's all.
So, just like always, you buy keys from one of the digital stores like Fanatical, green man gaming, Humble Bundle, etc.

Keys are the most common, but there are also gifts from Steam, and links to things like Humble, where you claim your key that way.

You add they key of the game when you create a giveway, and then it's automatically assigned to the winner chosen by RNG
During this phase, you can precise if the key is region-locked, if you wanna restrict to higher levels or specific groups.
Same in reverse, if you win a game, the key will appear automatically on your won page

You can still gift games from steam, but you can no more stock them in your inventory.
So if a sale ends before you could deliver the gift to your winner, you're in shit.
If you win a steam gift, the winner will try to add you on Steam to deliver it to you.

Welcome back and good luck for the future giveaways !

I didn't know that Steam sold keys themselves?

I'm fairly confident in saying that Steam never sold keys to users directly, the keys can only be requested by developers/publishers themselves and they then distribute the keys to 3rd party stores or sell them themselves. Specially because originally the whole key system was meant for activating the physical copies bought as disks, that's why they used to be called CD keys.

You're probably mixing it with the fact that some years ago we were able to buy games as gifts and keep them in our inventories to be sent at a later date, a function that Valve discontinued to curb cross-region trading. Although you can still get those kind of gifts when buying multi-packs from Steam, the extra copies show up as inventory gifts, but that only works for the games that have that option and there's only a few of those.

To summarise the SG experience:

1. Buy Steam keys from legit sites
2. Check for any Region restrictions or ensure key is Global
3. Host giveaway (Invite only/Group/Public) {easy}
4. Maintain an ok ratio {easy}
5. Participate in Discussions for fun or to seek Giveaways {easy}
6. Join Groups
7. Win Games
8. Play Games {easy}