I once got curious about that myself, so I downloaded some of the full size black and white TIFFs from the NASA site. (I'd link them, but I'm not sure where the archive is today.)
Anyway, if you bring up the shadows a long way in Photoshop in some of the pictures you'll find some little dots that look a whole lot like stars.
If you'd like to test this for yourself, take some photos of the full moon with a telephoto lens. The moon is extremely bright, as nighttime objects go, and the amount of exposure needed by the camera to capture a photo is short. Edit: The moon will be properly exposed, the sky dark.
There's something called the Looney 11 rule, https://en.wikipedia.org/wiki/Looney_11_rule. Basically, you can set the camera to these settings and obtain a reliably exposed photo of the moon. There's a similar rule for objects on earth which are in full sunlight called the Sunny 16 rule. The important thing to note is that there is one camera stop difference between these settings. Objects in full sunlight on earth reflect twice as much light as the sunlight reflected from the moon. And in terms of photography, that is a trivial difference.
Try those Looney 11 camera settings indoors sometime. There's not even enough light in your house for those settings to work. You'll see the lightbulbs, some reflections, light coming in through windows, etc, but not much else.
The important point to take away is the difference in the amount of light between the stars in the sky and the surface of the moon, is significant. It's large enough that it doesn't fit into the exposure latitude of color transparency film used (see https://en.wikipedia.org/wiki/Exposure_latitude), and is very deep in the shadows of negative film if it registers at all.
But rather than take anyone's word, this is something anyone who has a camera can try for themselves.
I am not sure this is the complete explanation.
On Earth we do not see stars because of light scattering (Rayleight and Mie) [0].
In the outer space there is no scattering and the light from each star represents a tiny point in the picture, but this point is bright. After all it comes from a sun. But the original image [1] has a dark background as well.
Maybe it is due to the fact that jpeg compression losses small artifacts or something else.
On some pictures it is possible to see tiny white points such as in [2]
We see stars during solar eclipses, so it is the scattering of the light of the sun that obscures them. It really _is_ the exposure issue, since the surface brightness of the Earth is that of full daylight and so full daytime exposures were needed. Stars will never show up unless longer exposures are used.
Here is an over exposed photo of Earth by Apollo 16 from the lunar surface, with stars visible:
Anyway, if you bring up the shadows a long way in Photoshop in some of the pictures you'll find some little dots that look a whole lot like stars.
If you'd like to test this for yourself, take some photos of the full moon with a telephoto lens. The moon is extremely bright, as nighttime objects go, and the amount of exposure needed by the camera to capture a photo is short. Edit: The moon will be properly exposed, the sky dark.
There's something called the Looney 11 rule, https://en.wikipedia.org/wiki/Looney_11_rule. Basically, you can set the camera to these settings and obtain a reliably exposed photo of the moon. There's a similar rule for objects on earth which are in full sunlight called the Sunny 16 rule. The important thing to note is that there is one camera stop difference between these settings. Objects in full sunlight on earth reflect twice as much light as the sunlight reflected from the moon. And in terms of photography, that is a trivial difference.
Try those Looney 11 camera settings indoors sometime. There's not even enough light in your house for those settings to work. You'll see the lightbulbs, some reflections, light coming in through windows, etc, but not much else.
The important point to take away is the difference in the amount of light between the stars in the sky and the surface of the moon, is significant. It's large enough that it doesn't fit into the exposure latitude of color transparency film used (see https://en.wikipedia.org/wiki/Exposure_latitude), and is very deep in the shadows of negative film if it registers at all.
But rather than take anyone's word, this is something anyone who has a camera can try for themselves.