Guess You are danish like me :-)
Your password is created on site by an algorithm made by a mix of your name, Your master password and the url. If You spell the url with capital you will get a totally different password for the site. it a kind of password-machine, and you will get the same password if you write the exact same information, and thats the secret about no server and no sync
The reason the website can calculate the same password as the app you're using has nothing to do with cloud, but everything to do with math.
Think of it this way:
5 + 3 = 8.
You don't need to save 8 in the cloud in order to do the sum again. You can buy a calculator in Spain, Japan or Brazil, and it will always say 5 + 3 = 8. You need to save nothing to do this math. The same is true for Master Password. <name> + <master password> + <site> = <password>.
Every time you want to calculate the password, you will need to type in your name and your master password so the algorithm can make the "sum".
In reality, it is of course not a sum but a cryptographic algorithm that protects against brute-force reversal attacks and all other known password and crypto attack vectors.
Help there is a picture on my phone that needs a master password. I do not know how to get one. Now on my smartphone there is a factory reset warning. What to do I only have 2 more turns to put in something