I use Extend Your Stay at the end of most cruises as its perk with zenith status.
You have to sign off like you do with the b2b process and hand in your sea pass card, you get a visitor card as a replacement for the time you remain on board. You also get wifi access for the duration and you can have soft drinks along with food from the Ocean view.
Your luggage will go off as normal (pit out the night before) and will be in a specific area in the terminal waiting for when you depart. They tend to like all those staying on to leave the ship by 2pm latest, and it's always a good idea to have a day bag if you chose to use the pools etc and then use the changing areas to shower etc prior to disembarking.
One thing I would stress is this is not currently available worldwide due to immigration issues and a lot of countries like the ship to have a "zero" count before starting the next cruise, therefore you have to get off the ship and only be allowed back on when boarding starts for the next cruise, by this time EYS is not worth it, and therefore not offered by the ship.
as others have already said, definitely don't miss out in Singapore.