There were fewer than 10 murders recorded in any one of the other years before 1659. In 1659 he recorded 70 and in 1660 he recorded 20. I note there is no spike in the number of executions in either year.
His table and methods are discussed from time to time among actuaries and statisticians, but this spike in murders seems to have escaped their commentary, which is odd to me as it is the number that really jumps out at me.
The spike might not indicate an actual rise in such incidents; but only a rise in awareness and/or incidents of report. Given that Graunt is a pioneer in demographic observations, we need to take his tables with a grain of salt. His methodology by modern standards may be lacking and though his actions are important for fields like mine own (criminology), they suffer from all the problems of being first in a newly developing field; the data collection is unrefined at best.
True policing would not develop until a further 200 years after Graunt's death, and even then was relatively reactionary: Which means that the police often only tended to apprehend perpetrators. Crime prevention was not a deliberate strategy, at least past simple deterrence and presence. This means and requires (a) that the crime already be committed and (b) that the Police are aware of it. The reporting of crime to authorities versus true crime rates are often heavily divergent. The case of the spikes in 1659 and 1660 could be as simple an answer of 'we were simply more aware of x amount of murders' and thus are given a truer indication of the rates, or that indeed there was a noticeable spike in the physical act.
Such things need to be viewed with a critical eye. Given the potential issues with his methodology, and this knowledge, most statisticians don't comment on this anomaly probably because its quite easily explained.