The key idea for the positive mass theorem is that asymptotically flat spacetime always has non-negative energy. Furthermore, of all spacetime which are asymptotically flat, empty Minkowski space is the only one which has zero energy.
This is an important result because it tells us that spacetimes such as Minkowski are inherently stable.
Now, the proof of this was lacking. How would one define 'mass' in spacetime? The ADM formulation of GR allowed you to do this. (You can look up: http://homepage.univie.ac.at/piotr.chrusciel/teaching/Energy/Energy.pdf and references therein) Soon after, there was the proof by Robert Schoen and Shing Tung Yau which is a nice exercise in differential geometry. Then came the famous Witten proof in which he constructed initial data on a Cauchy hypersurface for asymptotically flat spacetimes and showed that the energy density would always be non-negative. It should be mentioned that people know that supergravity theories had this positve mass spacetimes because the mass was defined in terms of squares of real numbers/ real, positive valued functions. Witten's paper is very readable and well articulated. (https://projecteuclid.org/euclid.cmp/1103919981 - Witten).
Following the work of Witten, there was an extension of the theorem to higher dimensional asymptotically flat spacetimes by Parker and Taubes (http://users.math.msu.edu/users/parker/Witten.pdf)
Theis inspired a host of papers, most famously the paper by Gibbons, Horowitz, Perry and Hawking for black holes (http://projecteuclid.org/euclid.cmp/1103922377)
There is still a huge amount of interest in the community regarding the stability of spacetime. For example, stability of vacua in string theory is a highly researched topic and so is the stability in dS spacetimes.