No matter what time the ADR computer or the CM on the phone tells you, your ADR is for the one and only Dessert Party of the night which begins 1 hour before the fireworks start. Get there 10-15 minutes ahead of time to check in at the podium. They give your entire party wristbands (because if you have to use the restroom you leave the actual party area and will need re-entry) and line you up. When the party officially opens they very quickly seat all the checked-in groups and then go back for the stragglers.
You will be shown to a table that literally has your name on it. You can then immediately go to the buffet, which is now up the ramp on the middle level of the Tomorrowland Terrace. The desserts and beverages are all self-serve. CMs do come clear empty plates from your table.
If you want to move up to the railing when the fireworks start, make sure to tell the CM in your area NOT to clear the table. That way you can have a final bite and go get one last drink refill before heading back out into the park. Waiting just 10 minutes really lets the gridlock on Main Street clear out.