Could someone briefly explain how these puzzles for giveaways work?

You have a question, you have an answer, leave them alone in a room, at first you will hear screams, don't worry, it is good, then you will see the question happy, and in few days her stomach will be huge and she will then born a link for the giveaway

You solve the puzzle and then enter the giveaway?

you figure out a 5 letter/digit code and then add it to the url http://www.steamgifts.com/giveaway but it's not always like that, just read what the poster has to say.

Also if you make a private giveaway only for me you can have your luck boosted by 200%

The puzzle maker gives a (usually) cryptic puzzle. You solve it. The clues usually give either the entire key for the game (less common) in which case it is first to solve it wins.
In the more common case the clues give 5 letters or numbers and these complete the URL link to a giveaway on the site. You'd substitute capital letters for the capital X, lowercase letters for small x and numbers for # in this URL http://www.steamgifts.com/giveaway/Xx#Xx

Puzzle: for example here
Find a solution => It will lead you to a private giveaway page. Puzzles often require brain.

^_^ they also have solutions u kno?

First of all, you have to understand the point of a puzzle. It's not just to entertain people, but it's also to reward the most astute amongst us as well as limit the amount of entries a giveaway gets. Telling you how to solve puzzles would defeat the purpose of the game.

Though, yeah, what the others said, you need to find a 5-digit alphanumeric code to input in the Steamgifts giveaway URL. Though good puzzle makers will always state that in their puzzle to accommodate for new interested users.

Oh and also, in 75% of the cases, Google is your best friend. If you have no clue, search for keywords related to the puzzle. Once you find out how to do certain puzzles, you will notice that many others simply recycle used ideas.