What keeps electrons on a negatively-charged conductor from leaving?
It is a quantum mechanical phenomenon. Wherever there exists an electric field potential there exist energy levels , i.e. stable orbital locations which can be occupied by an electron.
How does this happen?
Even the simple Hydrogen atom has a negative ion state, an anion. This is because the potential describing the hydrogen atom, neutral, still has many levels in which the electron can transition, and there exists a probability for an extra electron to be attracted to one of these levels, though one would have to solve the problem " two electrons one proton" energy levels. Even in the ground state the electron orbital has a p state probability, which means that the shape allows regions in space where the positive charge is open to attract other negative charges. These shapes are what allow the molecular bonding into H2.
In neutral atoms/molecules orbitals have shapes given by the complicated spin states necessary for the binding into molecules and solids, and there exist spill over forces that can be attractive to free electrons, i.e. have energy levels that are open to extra electrons. The stability then is guaranteed unless extra interactions occur, as described in the other answers.
One tends when speaking of bulk matter, as in conductors in the question, to forget the underlying quantum mechanical state, but if at the molecular and atomic level there do not exist stable quantum mechanical solutions with energy levels , then there would be no bindings. One speaks of surface tension and work functions etc to describe the many body effect of the underlying quantum mechanical state. The basic stability is due to the energy levels every time.