Think like this. 계세요 is the honorific form of 있어요. It means stay, exist, be in a place, and has a few usages that aren't pertinent to this discussion. You use 안녕히 계세요 when parting company with someone when that person isn't going anywhere. This would include situations where you are leaving their house and they are staying, when you're completing a Skype call with that person, or when you expect the person is reading what you are writing. Use 안녕히 가세요, or some version of that when the person is going somewhere as you part company. 가세요 is the honorific version of the word which means "go." So the simplest way to think, if the person is definitely going, use 가세요. If not, use 계세요.

안녕히 계세요 is said to the person who is staying and 안녕히 가세요 is to the person that is leaving. You would normally use it with people you are not close to or for more formal situations, like writing a letter. So if you were with someone and you had to leave first, you would say 안녕히 계세요 because they are staying, and they would say 안녕히 가세요 because you are going. For written situations I'm assuming that since the writer is the one ending the letter, they are technically leaving and the reader is staying, and thats why they end the letter with 계세요.