What precisely was the discovery facilitated at the end of Interstellar?

Near the end of Interstellar, (HUGE and nearly utter SPOILERS follow):

TARS, the robot, gathers data from the singularity that Cooper relays to Murph via Morse code on her watch. The outcome of this message allows for completion of the work that allowed for plan A, which, as in seen in the end, allows for a series of space stations to function as life rafts for the human race.

What was the discovery? What was the problem that needed to be solved?

Answer

Murph and Professor Brand were working on a technology to manipulate gravity.

During the events of the film, they are trying to come to a theoretical understanding which reconciles gravity with quantum mechanics -- the so-called Theory of Everything (although it is not referred to by this name in the film). When

Cooper and Murph first visit the NASA facility

there is a remark to the effect that the entire facility is designed to function as a spacecraft, if it can be lifted into orbit -- which requires control of gravity.

Gravity-manipulation technology would ultimately allow the creation and control of wormholes, like the one seen in the film. More immediately, it could be used to lift large numbers of people off the Earth into orbital habitats

like the one seen near Saturn at the end of the film. Presumably, these habitats would be safe from contamination by the blight which was destroying all food crops on Earth.

In reality, a Theory of Everything would be necessary in order to manipulate gravity and control wormholes, but it might not be sufficient. Interstellar takes a few liberties here, which is reasonable enough as it's a science fiction film, not a physics paper.