Creation of new nerve cells (neurons) in the adult human brain has only been only confirmed in a couple areas. (Granule cells of the dentate gyrus of the hippocampus, an area involved in forming new memories and cells which migrate to the olfactory bulb.) Much searching in other places including studies looking at uptake of radioactive particles from atmospheric nuclear bomb testing has shown that if new neurons are created in other areas, the rate must be extremely low. The plasticity of the adult nervous system comes from forming new connections between neurons, pruning existing connections, changing the strength of those connections and changing the sensitivity of a neuron to inputs. This forming of new connections and changing of connection strengths between neurons is lifelong and robust and is key to adult learning. Research that showed "critical periods" for certain kinds of brain development, such as binocular vision, led some to assume that this meant that new learning was more limited than it is and this idea sometimes got mixed in with the rarity of new neuron growth as a limit to plasticity. All that said, early intervention can be hugely helpful as the young brain is much more plastic than the adult brain.
The treatment regimen for the mice was to stop suppressing the gene that they had attached a chemically activated genetic switch to, so this particular treatment would not be applicable to humans who haven't been genetically engineered from birth to have this switch in the first place. If it were possible to insert a copy of this gene into humans with the proper regulation that might be a route for treatment, but the regulation bit is critical and difficult and I believe beyond what we can do with current technology. Perhaps someone who actually knows genetics might be able to comment on this question better than I.
They do suggest that there could be treatments for this genetic defect. They mention that drugs that block the (metabotropic) glutamate receptor "might hold promise for designing treatment strategies for non-syndromic autism," since the gene in question regulates this glutamate receptor.
The cool thing about the paper is that is showed that they could "rescue" the mice by turning on the gene switch and they were able to see that the mice recovered typical function at a cellular level and at a behavioral level, suggesting that if it were possible to re-enable this gene in humans that you might see some similar effects.
PS If you want to read the original paper, post here and I'll send you the pdf.