This theory does not explain the E-cat. It will require electric input to allow the electrons to merge with the protons forming the neutrons. So far as I know, the E-cat is requiring only heat.
Krivit gives a simpler explanation of Widom Larsen.
The key point is that it is different from BECNF in that it doesn't overcome the Coulomb barrier as BECNF does. Therefore, it would seem that the burden of scientific proof is higher for BECNF. Yet, Rossi's device is said to be operational. As far as I know, there is no device yet that employs the principles of Widom Larsen.
There are some similarities with the BECNF theory covered here previously. (see label on left sidebar for a list of these posts) Both theories posit that quasiparticles allow unusual events to occur.
In the case of Widom Larsen, low energy neutrons are formed. The neutrons can be absorbed by nickel nucleii and release energy due to beta decay.
In the case of BECNF, the Coulomb barrier gets suppressed by the Bose Einstein condensation. This allows close enough proximity of ordinarily repelling like electrical charges- protons and nickel nucleii- so that they can fuse and release energy.
These unusual events lead to the production of heat which explains the "cold fusion" phenomenon. It is important to clarify though, that Widom Larsen theory is definitely not fusion.
Both theories rely on the many body theories of quantum mechanics.