Why use this ? Because you don't trust sites that retain your password, because you are not able to remember more than one password whereas you should use a different password on each site you visit.
Is this site secure ? This site does not use external resources, nor submit forms to a distant server. You can even disconnect your computer from the network when this page is loaded if you don't trust me (and you shouldn't...)
How to generate a password ? Choose a long master passphrase that you will remember, enter the application name or the server name (e.g. "facebook", "foe.bar.com"), then the login used on this site ("firstname.lastname@example.org", "root") and copy/paste the password generated for you. If you enter again the same values, you will get the same password. So remember your master passphrase well, do not write it down, do not use it as a password on any website, in one word : keep it secure !
What kind of passwords does it generate ? The passwords generated by this site contain 12 characters, with at least 1 lowercase letter, 1 uppercase letter, 1 numeric and 1 special character.
What if this site disappears ? Well, this site is made of this sole page, I encourage everyone to copy it on other web servers. This is the only but affordable guarantee that you will still be able to generate your passwords in the future.