A visit to my local shoe shop tells me this is not a widely accepted claim. Nonetheless, the claim is notable, and appears on many lifestyle blogs. The linked site offers some evidence that flat feet are caused by or related to the wearing of shoes, but no evidence that wearing minimalist shoes or going barefoot will fix the problem.
It is not obvious to me which answer is more plausible:
Wearing shoes with lots of support means some of the muscles or tendons in your feet don't get regularly exercised (the shoe does the work). This systematic weakness leaves you with injury prone feet. Switching to shoes without support causes these muscles or tendons to be used again, strengthening your feet.
Foot structure is determined by factors that are difficult to change (like bone lengths), and will not be changed significantly by your choice of footware.
There seems to be some indication that wearing minimalist shoes does not produce a change in leg strength, but does increase the rate of bone injuries to one's feet. However, as mentioned in this Skeptics.SE question, there doesn't seem to be evidence that regular running shoes provide benefits at all.
What I'd like to find is a proper study assessing changes in the strength or shape of feet following the adoption of minimalist shoes.