You steal energy from the object you're slingshotting off. The black holes are slowing (infinitesimally) down every time you do it, gifting their energy to the accelerated object.
Hence the article talking about potentially detecting this sort of behavior by monitoring for prematurely merging and overly eccentric binary black holes.
https://en.wikipedia.org/wiki/Gravity_assist
You steal energy from the object you're slingshotting off. The black holes are slowing (infinitesimally) down every time you do it, gifting their energy to the accelerated object.
Hence the article talking about potentially detecting this sort of behavior by monitoring for prematurely merging and overly eccentric binary black holes.