The mirror bounces light up and onto the "focusing screen" -- that's the thing you see on the top of the chamber in the camera. The image actually comes into sharp focus on that particular spot because it happens to be the same distance from the back of the lens to the sensor as it is from the back of the lens to the reflex mirror, and up to the focusing screen. (Also, when you adjust the diopter wheel next to your viewfinder to focus it... what you're really doing is helping your eyes focus an image on that screen.
On some cameras that screen is replaceable. My 5D II has user-replaceable focusing screens (e.g. I can get screens with grid-lines, screens without gridlines, screens with extra-fine frosted effect to exaggerate the blur of out-of-focus areas (intended to help with manual focus, etc.) My 5D III does not have replaceable screen (well... not designed to be replaceable, I think some aftermarket companies found a way to replace it anyway.)
But there are two reasons that come to mind for why you should never touch that screen...
#1... one side of the screen is frosted, the other side is shiny. If you touch the frosted side, you'll smudge it... since it's not a shiny surface (unlike the mirror or sensor filters) you'll have a REALLY hard time ever getting oils off that side of the screen and will almost certainly damage it trying.
#2... the light meter is technically up above that screen. The amount of light transmission through the screen is critical to the accuracy of the metering system. I have two different focusing screens for my 5D II. they both "look" the same and if I had to guess, I'd think that they both transmit about the same amount of light... it turns out they do NOT. The camera has a menu setting and I have to tell it which focusing screen is in the camera. If I don't set it correctly, the camera will incorrectly meter all my shots. In other words... if you damage that frosted coating, even if the screen "looks" ok, it can still have an adverse affect on your photography.
And then there's the reasons everyone else has already stated... once the reflex mirror swings clear, the light path goes from the back of your lens directly to the image sensor and any dust on the reflex mirror or focusing screen can't effect your shot. If the dust doesn't come free with a few puffs of an air blower, I wouldn't try to use anything more aggressive (unless you're prepared to have the focusing screen replaced.)