They surmise that the palladium catalyst is somehow bringing the atoms of deuterium close enough for them to fuse at normal temperatures. Such a process could occur, and is known as tunnelling or barrier penetration, a quantum phenomenon which involves travel at the speed of light through the barrier. Contrary to popular belief, this is not forbidden, but the probability of this occurring decreases extremely rapidly as the barrier width is increased. The energy barrier between the positively charged nuclei would still be excessively high no matter how close the palladium could bring the deuterium atoms together so the probability for fusion to occur is extremely low. Heisenbergs' Uncertainty Principle ensures that this does not mean it does not occur, but it will be at a very low, hardly detectable, rate.
If nuclear fusion was occurring, then gamma rays characteristic of fusion between deuterium nuclei should be emitted, and this has not been observed. The released heat spectacle is most likely explained by a low energy chemical rather than a nuclear reaction. Sorry to pour cold heavy water on this dream.