How was humanity able to use the quantum data that Murph received from her dad's watch in Interstellar?
Dr. Brand, had already solved the necessary equation, which was required for the
question of how humans can escape Earth's gravitational pull en masse.
But he needed some more data from a singularity behind a black hole to complete it. All this was needed for safe migration of all humans from earth to the other planets, and this indeed was PLAN-A.
Cooper gets this data from TARS and translates it to Murph, who successfully completes the equation. (Murph is seen tossing the papers in joy.)
And yes, humanity able to use the received quantum data. While an exodus is not shown, it clearly evident from the last scene where a new human colony has been established, which is currently revolving Saturn (as stated by the doctor who attends a recovering Cooper.) Dr. Brand's daughter also succeeds in reaching another planet which can sustain life, but till the movie's end, she was the only one on that planet.
It enabled Murph to complete Professor Brand's equation which was required to enact "Plan A" for the evacuation of Earth.
Brand and Murph had previously been unable to complete this equation as such quantum data was only available from a black hole. TARS going into a black hole and transmitting this data back to Cooper in the 4th dimension, who then tranmitted it to Earth via the watch, allowed for its completion.
EDIT: Updated answer above - it was actually TARS, not Cooper, who got the quantum data, as pointed out by ed0.
"It enabled Murph to complete Professor Brand's equation which was required to enact "Plan A" for the evacuation of Earth"
That's right, but but the one ending up inside the black hole (in order to see the singularity and acquire the quantum data) is TARS the robot, who contact Cooper while he is trapped in the 4° dimension, through which he transmits the data to Murph. In the end, she is able to complete the equation and evacuate humanity from the Earth
1It's a shame this is getting downvoted, as it's a good point which I've now incorporated into my answer, but perhaps it would have been better as a comment than a separate answer. Nov 16, 2014 at 13:41
Where is shown that the Earth was evacuated? I already asked this in another comment here...– LincolnNov 17, 2014 at 21:45