It means you are staying at home. PTS is the Navy's way of keeping people that are in. They may have to work out of rate. Or change ratings depending on manning. What I'm figuring is your guy was told take these orders, (that suck) or get out.
An "unaccompanied tour" means the sailor will be stationed in a place where he or she cannot bring his/her spouse or children, usually for 2-3 years. This is common for low-ranking sailors posted to Japan, UAE, Diego Garcia, or some other overseas locations. The spouse/children can remain at the previous military base, or can move "home" to be with family while their sailor is posted overseas.
PTS is Perform To Serve. Quite simply, only the best sailors are offered the option of reenlisting. The Navy is too full right now, too many folks at the top aren't leaving the Navy because there aren't any jobs on the outside. So only the very, very best are kept. The rest are "laid off" at the end of their 4 or 6 year commitment (whichever they signed up for).