It stands for allowing someone to continue doing or use something that is normally no longer permitted (due to changing regulations, internal rules etc.)
It means to give benefits to old users that you have stopped giving to new users. For instance if your online service is completely free and then you restrict certain features for new users, you might grandfather old users into the premium plan to keep them happy (since it was free for them initially)
To grandfather a subscriber is to keep them on a dormant (retired) plan that is no longer on sale to new customers. Generally it is not worth the disruption of forcing old (grandfather) customers to switch only a new deal, but better to let them continue paying at the same old rate.