Does Memcomputing really solve an NP-complete problem?
I feel this has been answered sufficiently in the comments, so to just sum everything up:


The authors do not claim P=NP, which is a statement about deterministic and nondeterministic Turing machines.
The authors propose a model of computation that they claim to show is equivalent in power to nondeterministic Turing machines.
The authors construct physical machines that implement this model of computation for small input sizes.
The authors argue that building larger versions is physically realizable/possible with polynomial-sized resources.
This last claim, which is of course not proven and not really a formal statement, would imply a that it is generally physically possible to solve NP-complete problems with polynomial-sized resources.
Scott Aaronson in a blog post explains why this last claim is problematic and why the scalability of their approach has problems: http://www.scottaaronson.com/blog/?p=2212