Nearby Mar Lodge Estate holds a Gathering Ceilidh on the Friday before the games and you might want to keep an eye on their website and Facebook page nearer the time to get updated information. This year tickets were £20 each and this included stovies and there was a licensed cash bar.
Mar Lodge is about four miles away from Braemar so you would need to drive there from the village, or contact Mar Lodge to see if any transport is on offer. There might be a shuttle bus, but you'll have to reserach that. We have very strict drink/drive laws in Scotland now and it is not worth running the risk. There's no public transport link to Mar Lodge.
On the Sunday after the gathering you could drive up to Crathie Kirk to attend the service with any members of the Royal Family who will be there. Or just stand outside and watch the comings and goings. If you attend the service the building has been designed in such a way that the royal pews are not really visible from the main body of the kirk. You might see someone's hat depending on where you are, but that will be as much as you get.
Other than that I'm not sure much happens in Braemar the day after the gathering because there are too many people nursing sore heads :-)
However there are plenty of walks to take and other things to do in the area, including a visit to the Royal Lochnagar Distillery if that appeals.
I'm not that far from Braemar myself and very close to Crathie Kirk, so if you have any further questions or if there's anything else you want to know, I'm happy to help. You can send me a private message if you wish.