There's a quiet pressure put on us often to care, about anyone and everything, and that's neither a fair nor realistic expectation. Not caring doesn't necessarily make one bad, cold, or insensitive. If we're being honest with ourselves, we'll see that to care, truly care, from within, calls forth significant amounts of energy. Energy which is ever renewable, to be sure, but not always plentiful (we all have our peaks, as well as our plains and plateaus). Wouldn't it be a wiser use of our energy to admit this, and give ourselves permission to not care when it's OK? (Loosely translated, "not caring" in this sense essentially means "not thinking about".)
Don't care what's going on in my life? Alright with me; much better than pretending you do, in my view. It's a question of context. In certain situations – for instance, when someone treats you unjustly, or attempts to insult you – it becomes beneficial to not care. The more you act like who you are not, or act contrary to how you actually feel, the more opportunities for misunderstandings and manipulations increase.
Let's say someone on the job doesn't like you, for reasons unknown to you. That's fine; really. We all think a little differently. When you factor in all the nuances and variables at work in biological chemistry, all things – including people – may not agree. By accepting this – in my experience at least – it becomes much easier to work together as you accept them, and the particular dynamics of the work relationship, for what it is. No more, and no less.