Warming of the Earth doesn't happen instantaneously. After emitting some CO2 (or other greenhouse gas) into the atmosphere, it takes quite some time before the Earth finishes warming up.
Right now, the Earth has experienced a little more than half of the warming that will eventually result from the present concentration of CO2. That basically means that if we stop emitting all CO2 and other greenhouse gasses tomorrow, the Earth would very likely still hit 1.5C of warming.
But the authors are very much correct that the 2C goal isn't safe by any means. Current model estimates are that the ice sheet of Greenland will destabilize and completely melt at somewhere close to 2C of warming. That would cause approximately 7 meters of sea level rise, in addition to all of the other sources of sea level rise (note: it'd take a few hundred years for Greenland to melt, but even relatively small amounts of sea level rise can be devastating). There's a fair amount of uncertainty here. Greenland might not destabilize until warming of 2.5C. Or it might start occur right at 2C. It would be far, far better to never find out.
2C is primarily a goal just because reasonable and cost-effective but aggressive measures to limit greenhouse gas emissions can (at present) allow us to reach a 2C target. We really wouldn't have to sacrifice much of anything to reach a 2C target (and the extra construction of renewable energy infrastructure would be a boon to many economies who are currently struggling). But we would have to overturn the massive influence of the fossil fuel lobby both on government and on the public discourse.