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So I've been trying (with minimal success) to find any quality information on this, but there are a few bits and pieces out there.
My understanding is that the most likely theory here is that there's a low energy mechanism for generating neutrons in condensed matter via 'heavy' electrons (high effective mass due to lattice phenomena), and that these neutrons can be used to trigger energy producing reactions (there's a lithium based cycle with no net consumption of lithium, for example). The reactions themselves aren't new, but producing neutrons cheaply enough to generate a net energy gain is. I don't have enough of an understanding of the theory to really judge how feasible it is, but the idea that electrons in lattices can behave in interesting ways (c.f. superconductivity) isn't crazy enough IMO to dismiss the idea outright.
I think this is relatively orthogonal to Rossi and Co., although I believe there was some interaction between him and NASA at some stage. He's definitely mishandled the public relations around his announcement, is likely out of his depth in terms of understanding what he's doing and may well be attempting fraud. That doesn't change the fact that there may be some worthwhile science to be done in the field.