How do I know I've won a giveaway?

You'll be notified via the website and email.

there would be popup message informing you about winning the game

Also, on the top-right menú, you'll see three icons: a gift box, a trophy, and an envelope. Any number on those means there's a giveaway you've made that has finished, a giveaway you won, or a message in a comment, respectively. So if you see a number in the trophy icon, you've won something. A pulsating number means the key is available to you (tho you can also receive a gift link, or maybe you have to get to be friends with the giveaway maker in Steam to receive the gift directly, there are several ways to get your won game).

But yes, basically you'll get a pop-up with the space cat trembling in excitement telling you that you won a giveaway (patrons may have customized animations instead of the spacecat.)

You only get an email if you have it enabled in the settings.

I think it's possible you won't notice you won a giveaway if you have emails disabled and you haven't visited the website for more than a week, because after a week of no feedback givers are allowed to reroll which I'm pretty sure resets the trophy icon.