I’ve usually thought of that gesture as a reflexive instinct to prevent vomiting. You will see people cover their mouth when they see shocking scenes that could induce nausea or disgusting sights that could induce revulsion.
If you watch some of the “2 girls 1 cup” reaction videos on youtube, a huge proportion of people cover their mouths and it does suggest it’s an anti-vomit thing. 
Here’s a great example: http ://www.youtube.com/watch?v=UISZTxKJjkg (this is safe for work, but the title alone might be enough to cause problems so link is broken!)
After 30s you get the reaction - not only is it hilarious, but you can see both lads cannot keep their hands away from their mouths.
And in this clip - http: //www.youtube.com/watch?v=OBZt1JUmrRo&feature=related - eventually all three girls have covered their mouths in horror, likewise in this one http: //www.youtube.com/watch?v=m_ZB0-_FU_U&feature=related
It’s how people silence their silent scream, if that makes any sense.
I knew the thread would be about that picture as soon as I saw the title, I asked myself exactly the same question when I saw it. Personally I think it’s a combination of the learned behavior of not showing your open mouth for politeness sake, and an instinctive comfort action, like crossing your arms - it makes you feel a bit more ‘protected’ and gives you something to do with your hands. It probably has the added benefit of helping you to stop yourself making any noise if the situation requires it.
Heh.
When you open your mouth, there is a slight chance that your soul will escape. So, the tendency to cover your open mouth gives a slight survival advantage. Over millions of years, covering your mouth is naturally selected and we all have much safer souls.
Weird, this internet is.
Saw that photo on the news of the royals covering their mouth, googled ‘humans covering mouth shock’ and ended up here.
Kinda like that guy on Close Encounters who builds that mountain out of mashed potatoes and discovers other people doing the same thing…
Welcome to the fun, Yoda. 
Not sure if it is connected but heard a (psych) anecdote that demonstrates whether people are listening, feeling or seeing centred. You tell a horrid, cringeworthy story and different people will cover different parts of their bodies at the worst parts of the story. Some will cover their mouths, some their ears and others their eyes.
Apologies for the hijack if this is totally the wrong tangent.
Funny I was going to start this same thread when I saw this picture (scroll down to the second photo):
http://news.sky.com/skynews/Home/World-News/Car-Careers-Into-Crowd-Watching-Dutch-Queen-Beatrixs-Royal-Motorcade-in-Apeldoorn-Holland/Article/200904415272530?lpos=World_News_Carousel_Region_1&lid=ARTICLE_15272530_Car_Careers_Into_Crowd_Watching_Dutch_Queen_Beatrixs_Royal_Motorcade_in_Apeldoorn%2C_Holland, but I was going for the angle that it seems to be women only who do this (the photo in the OP notwithstanding - must be a girly man :p). I think Duck Quack’s Echo’s suggestion of the anti-vomiting reflex has merit, but I wonder why this is so much more prevalent in women?