There are a bunch of good reasons why you should try to learn to sing while you play:
Now, if you're one of those who would protest, "But I can't sing," let us stop you right there. Yes you can.
Before you start trying to sing and play at the same time, you have to know that you can sing in the first place. Which you can, even if it's only a little. Everyone can sing. Everyone should sing. Singing is good for you. Granted, not everyone has the voice of an angel (or Bono, anyway), and there are bound to be a few who just couldn't carry a tune even if it had handles. For our purposes, however, you're not one of them. You can sing a song.
To do so, you don't have to have a great voice—you just have to have a little control over the voice that you have. Rock and pop are full of successful singers who don't have technically stellar singing voices; just look at David Lee Roth (who we love to pieces). Most of them, in fact, are not formally trained singers. But that never stopped them, so don't let it stop you. It is true that singing is an innate talent for a relative few, but for everybody, singing is also a learned craft and skill that can be worked on and improved. Greats vocalists such as Sinatra, Sting and Pavarotti were all naturally gifted singers in the first place, but they had to work on it, too.
The best way to get a little control over your singing voice is to just start doing it. Start singing. Imitate your influences. Consciously dissect the material you like and see how the range, phrasing and timing work. Pay attention to what you're doing, and you'll discover what your vocal strengths and weaknesses are.
To reiterate, you can sing. And if you can sing by yourself, you can sing while you're playing. Again, the best way to learn to do it is to just start doing it.
Look at it this way—you didn't learn to ride a bike by sitting around studying and reading about how to ride a bike. You learned by getting on a bike and doing it. You started slowly, you saw other people riding bikes, you had help, and maybe you had training wheels for a little while, but you learned to ride a bike by riding a bike. It was shaky at first and you fell a couple times, but you kept at it and pretty soon it was as natural as walking (which, it bears remembering, you also had to learn).
Start slow. And start simple—you might want to try something a bit more modest than "Bohemian Rhapsody" right off the bat. Again, imitate your influences. Splitting your attention between what you're playing and what you're singing can be tricky at first, but it does get easier and it can quickly become second nature. And once you've got it, you've got it for good—just like riding a bike, you can't really forget how to do it. You might get rusty if you don't keep doing it, but you'll never unlearn it. You might even find that you enjoy making music even more once you've learned to sing while playing.
So grab you guitar and give it a shot. You can do it. We'll even count you in: And a one, and a two ...