The authors present many potential explanations that were put forward at that time to explain the phenomena. These included, more cops on the ground, stringent sentences, broken-window policy and so on. In the author's assessment all these hypotheses turned out to be wrong.
According to the author's analysis the likely cause was the ratification of the Roe-Wade judgement.
The causal chain is as follows, legalized abortion resulted in the reduction of the number of people who were likely to get very little parental support and social support while growing up. Arguably, this population is more predisposed towards crime and its relative reduction led to a fall that aligned with the timeline -- around the time the newborns would have hit the late teens.
Attributing an irrefutable cause is hard.