There is one big advantage that the air outside the house has over the air inside the house, and it is this: On a sunny day, the air outside the house comes with a good dose of ultraviolet radiation from the sun. Ultraviolet rays, it turns out, are good disinfectants -- they kill bacteria. So if your mother has you haul everything to a sunny spot on the driveway so that it gets several hours of strong sunlight, she is using the sun to disinfect things. With something big like a mattress or a rug, it's one of the few inexpensive ways to kill bacteria -- you can't exactly throw a mattress in the washer!
It turns out that ultraviolet light gets used as a disinfectant in a number of different products. For example, you can buy swimming pool systems that disinfect the water in the pool by running it past a strong UV light. Some municipalities do the same thing to augment the chlorine that treats drinking water. You can even buy a device that uses UV light to disinfect your toothbrush!
In third-world countries, there have been experiments using clear plastic soda bottles filled with contaminated drinking water. People leave the bottles in the sun and it actually helps to reduce bacteria in the water. See, for example, this page.
There might be other reasons your mother has as well. If you go back in time a century or so, there were no vacuum cleaners. You hauled the rugs outside to beat the dirt out of them. Even with vacuum cleaners, they usually don't get everything, so beating the rugs helps to control dust inside. Moving the mattresses outside may make it easier to get under the bed to sweep there. Or it may just be an annual tradition for her. In any case, the UV light that everything receives can't hurt!
Here are several interesting links: