Mesenchymal stem cell therapy is a promising and scientifically proven cell therapy.
This article explains in detail the types of Mesenchymal stem cell therapies to treat peripheral neuropathy.
Peripheral nerves have a limited regenerative potential, meaning if peripheral nerves are damaged it is difficult for healing to occur.
(E.g. As opposed to the liver, a highly regenerative organ - "51% of the original liver mass can regenerate back to its full size".)
Therefore peripheral nerves are harder to repair when compared to other tissue.
Upon peripheral nerve injury, molecular disease events occur whereby multiple cell types assist each other in order to restore nerve function.
The Image below shows the cell types involved and the stages of nerve injury.
For this explanation we need to focus on stage 4. Schwanncell alignment; axon regeneration.
Schwann cells change their function when the nerve is injured. Originally their role was to support the nerve function & insulate conduction along the nerve. Now their function is to grow, align and join the gap/site of injury. We can observe non-myelinating Schwann Cells perform this function in stage 4. Schwann Cells direct neuron repair & secrete repairing molecules (neurotrophic factors).
(Interestingly axon regeneration occurs at ~1 mm/day).
Mesenchymal stem cells are easy to obtain, located in fat (Or hard to obtain in bone marrow). Mesenchymal stem cells are compatible with the person they are derived from (Low risk of rejection). They have the ability to turn into a cell type similar to that of Schwann Cells, which can release the repairing molecules aka neurotrophic factors - as refereed to in the paragraph above.
Mesenchymal stem cells can turn into Schwann Cells in a petri dish and have been shown to also turn into Schwann Cells at the site of injury.
So to answer your question yes, I understand your skepticism - as for some claims are false. But this claim has merit.
Edit: To make this answer not an 'only link' answer.