First, I LOVE Scrivener. Love. LOVE! However, I’m having a little trouble upgrading. I paid my $25 on the website. I did the update thing in Scrivener 3 times - nothing ever re-launched and when I re-launched it myself, it was still v.1.54.
Open the dmg, drag the icon to your programs folder and replace the old Scrivener version.
When you first launch Scrivener 2 it will ask you for your new licence number.
Note that every time you open an old project for the first time Scrivener will inform you that it will have to update the project. Before doing so it will backup the old project so even in the case that you want to go back to 1.5—you will never ever want to go back!—your old projects are intact.
Everything above is good. The program won’t upgrade to a major version from within the old version for the sake of not confusing people too much if they don’t read the update text and are suddenly “locked out” and needing to pay, with no clear of way downgrading. So yes, you need to download from the website to get your new copy running.