I am not a physics geek, and am afraid to say that I don't get it.
The limit in question is a certain mass and seems to address whether a star will become a white dwarf, or alternatively, a neutron star or a black hole. The behavior of electrons as the aging star collapses into an incredibly dense form is key. Above this limit, there is sufficient mass to overcome the electron forces.
The bet described is whether a certain size blackhole will be found to exist as described.
Several of the world's leading physicist engage in bets involving their pet theories. Several of these bets are documented as shown and the documents are posted on the campus at Caltech in Pasadena, California. Some past bets have been resolved and the loser documents his concession on the document and has to pay the bet.
For people at Caltech, it is more interesting than betting on the Super Bowl.