Nothing spectacular (like time travel) happens when two black holes merge. When two black holes merge, they just form a big black hole, that is all. Of course, there ARE some interesting processes that happen before the merger and that is the emission of gravitational waves. When two black holes orbit each other, gravitational waves are emitted in accordance with the laws of general relativity, and as a result the black holes come closer to one another. Finally, there is a burst of gravitational radiation close to the merger. These are the kind of events that scientists are trying to detect with their gravitational wave detectors like LIGO or pulsar timing arrays like NANOGrav, to prove another facet of general relativity.