The black hole would not be as dangerous or hard to handle as a massive quantity of antimatter - the commonly prescribed fuel for starship propulsion. Confining antimatter is a big problem. One leak and kaboom!
But a black hole confines itself. What's more, simply making a black hole drive would require millions of times less energy than synthesizing and storing a comparable amount of antimatter.
There is any number of ways to tap the energy of the subatomic black hole to make a stardrive. High-energy gamma rays gushing from the micro black hole could be converted into electrons and positron pairs. These particles would be directed by electromagnetic fields into a collimated jet.
The black hole stardrive would have to last long enough for the interstellar mission and not evaporate away, have a mass comparable to that of the starship it is propelling, and yet be powerful enough to accelerate an exhaust to a reasonable fraction of the speed of light.
The authors dismiss the viability of other proposed stardrives based on current physics. An interstellar ramjet that sucks up tenuous hydrogen produces more drag than thrust. My favorite, laser beam propulsion transmitted from the home star, runs into the problem that the beam spreads too fast.