Unfortunately I don't have access to the paywalled PRL paper, but I think Bert Barrois's comment sounds right, so I'll go ahead and post a possible explanation as an answer. Maybe others who have access to the article, or who can find other information, can provide better answers.
The Q value of this decay is unusually low at 2.5 keV (which is presumably the reason for the very long half-life). More typical beta decay energies would be in the MeV range. The energy of the K shell in osmium is 74 keV, and the less tightly bound orbitals would go down in energy from there to whatever the ionization energy of osmium is, presumably on the order of 1 eV.
So if 187Re tries to emit a beta with an energy in the range from 0 to 2.5 keV, the beta is emitted with an energy that is sort of in the middle of the range of energy levels occupied by the electron cloud. Although this is just a hand-waving argument, it does seem plausible, as Bert Barrois suggests, that this frustrates the decay due to the exclusion principle.
Why does this happen, and is this only characteristic to rhenium 187, or all beta decaying nuclei?
Assuming this explanation is right, then it's a mechanism that would apply only in very unusual cases. In most cases, beta decay energies are on the order of MeV, which is at least one order of magnitude greater than the K-shell energies, even in heavy elements.