The narrator is recalling an age of co-existing extremes. It was the best and worst of times, a time of hope and despair, etc. England and France are both ruled by monarchies. Treaties have long been settled. It is the year 1775.
England turns its interests to spiritual matters, though it keeps an ear out for news coming from America. France concerns itself with making and spending money. Horrific punishments for small crimes are being sentenced to the condemned. There are those working silently towards Revolution, for speaking aloud would brand them traitors.
England is being plagued by crime, mainly burglaries in the cities and highways. Carrying the mail is a risky job, and often it is waylaid. The gallows are busy, and make no distincion between a petty thief and a murderer.