How does material get swallowed and torn apart by a black hole and have its light vanish. But somehow the same material escapes years later. After its light couldn't even escape?
They watched it for a long time, too. So how does material escape after its light can't even get away to show its glow they detected?
Anything that passes into the event horizon of an astronomical black hole is gone from our observations. It won't be seen again.
But material can orbit a black hole without falling in. If there is sufficient material, it can be heated by interactions with other material and glow brightly. As this glow is from the outside, we can observe it without the problem of something escaping the black hole.
There was a recent news article about a black hole interacting with a star. This star was being disrupted by tidal interactions. But just like the other things we see, it wasn't actually falling all the way into the black hole. Just getting close in the orbit around it was enough to cause visible disruptions that were seen.
You are talking about Hawking radiation, which makes black holes evaporate. While General Relativity explains how BHs form through gravitational collapse, it's quantum mechanics that explains their demise. Vacuum fluctuations cause the creation of particle-antiparticle pairs near the horizon. Normally these would recombine almost instantaneously, but in this case, it is possible that one of the pair crosses the horizon. In that case the other can escape, and it has the effect that the BH has lost some mass.
For stellar-mass or larger BHs, this happens extremely slowly. For a solar-mass black hole, it takes about $10^{67}$ years. For comparison, the universe is only about $10^{10}$ years old. However, as the BH loses mass, the process goes faster and faster, and the final stages happen explosively.
Note that this process cannot actually happen at present, as the temperature of a stellar BH (about $60 nK$) is far below the CMB temperature of $2.7K$. It will only start when the universe has expanded far enough and its temperature has dropped below that of the BH. It does occur at present for BHs with a mass of less than about 1% of the earth. BHs like that may exist as remnants from the Big Bang. These have been proposed as a possible candidate for Dark Matter. However, if they exist, we should be able to see the explosions as some of them go through final evaporations. No such explosions have been observed.
