So, you've found your fish on the site and its beautiful. Assuming you've made an account (it's free), you can bid on the fish. I won't go into detail about bidding right now, but bid what you feel comfortable or use the "buy now" option if there is one. If/when you win, you will have time to pay the breeder or pay right away; this option is best lest you forfeit for non-payment (and get a bad rating...). The breeder will charge you their shipping fee (which is usually quite reasonable). This shipping cost is the cost from the breeder to the transshipper in your country.
Once that is all set and the breeder ships to the transshipper, it'll be fully out of their hands and fully into yours. They sold you the product and now it's done! Your transshipper (literally an import agent) is in control of your item. It is wise to read up on the people in your area; generally, you can choose who you want. These folks tend to be spread across a region and it makes the most sense to choose someone closest to you or to a major shipping hub/city.
Typically, you tell the breeder who you want to ship to, get in contact with them, all that. Breeders usually have transshipper info at the bottom of their auction listing.
Now you've paid your fish's price and the breeder's shipping fee. Your transshipper will get in touch and tell you their price; as the fish will then be (typically) unboxed, allowed to rest in a small tank, and then re-boxed up to get to you, this price can be a little steeper than you may expect. It's a good idea to get the quickest shipping speed possible from your transshipper.
The transshipper's fee is paid through whatever means they accept (usually Paypal) and they ship based on weather, fish health, and payment clearing. They usually have their own pages that tell you their policies, so make sure you look it up!
Your fish will then get to you, and tada! You're set.
Here's a scenario:
A black halfmoon male is for sale from seller ThailandFish4u for $20 auction or $25 Buy Now. You elect to save some time and Buy Now. The fish, based on the seller's listed price, will be $5 to ship from him.
You pay these two fees and are out $30.
You get in contact with the breeder and choose a transshipper nearest your area named Jane Doe. Jane Doe is contacted, things are arranged, and you pay her shipping and handling fee of $35.
You wait a number of weeks, because it takes time to pack the fish in Thailand, ship it, get it in your country, have the shipper let it recover, and box it up again. In total, you've spent $65 when you get the fish.
And that's it! It's fairly simple. Most Thai breeders have FAQ pages (like this one) and most shippers have their own websites. Language barriers, lack of communication, and more are all issues that may crop up. Your fish may die and you'll only get back the initial $25. It's up in the air: are you willing to play this game of chance? If it makes you nervous or you do not like to deal with people business-wise, buying in your area/country is a better option.
Hopefully that quick spiel helps! When in doubt, ask your seller or shipper! :)