That might sound pretty obvious, but really understanding that means not putting them on a pedestal, not having grand expectations of who they are going to be as a person, and realizing that they have work to do, goals to accomplish, and fears and emotions just like anyone else. That kind of understanding will make you a better professional, better at working with musicians, and less likely to be the intern who gets fired for crying when a big star visits the office (I actually heard a story similar to this, so it isn't far fetched). Even if you believe your favorite star's music is perfect, it doesn't mean that they are. Don't react with disappointment if they aren't what you expect, even if you think what you are saying is a compliment (ie. don't ever say anything like "you're so much prettier in person" or "you're a lot nicer than I thought you'd be"). In fact, the day-to-day life problems of your celebrity clients are more likely to happen in the public eye and potentially make them even more stressed out (would you want your bad hair day to be a news story?). Don't judge if you work with someone who might have a habit of turning up in the tabloids (especially if you're only an intern and you're not working in PR).
The bottom line is that if you find yourself working with celebrities as part of a career in entertainment, you need to remember that you are there to work and so are the celebrities you are working are with. You should treat them with respect, a sense of privacy (as in don't tweet your friends saying who is in your office) and professionalism, just as you would any other co-worker or client.