You should get an email a day or two in advance letting you know the booking details, including the link you will use. For Animal Kingdom, booking opens February 1, and for Epcot, booking opens February 15.
Bookings have been starting at 11 am Eastern time. Sometime before 11 am ET, you should click on the link. You will see a countdown clock. You don't need to do anything other than keep that window open. When booking begins, you should see a queue number (randomly generated among everyone logged in by 11 am ET). When your number is reached, you will be able to enter your DVC member number and sign up for Moonight Magic along with everyone on your reservation.
You can use the link to open windows on multiple devices before booking begins, then close other windows after you get in on one device. You should not, however, refresh your window -- that will generate a new, later queue number.
If you are unable to book because your queue number isn't reached before bookings close on "Early Registration" day, you can try again in the "General" registration the following week.
This was actually far less complicated than I expected. For the Animal Kingdom Moonlight Magic in December, I was fully booked by 11:10 am.
