The sentence means that life like 19th century literature (stories or novels) is full of coincidences that most people would find difficult to believe are possible. For example if you are at a place and your best friend happens to be in the same place at the same time...this is a coincidence. These sort of events often happened in 19th century books because it was necessary for the plot and the use of coincidence was a normal technique for 19th century writers. Charles Dickens is famous for his use of coincidence. Modern literature tends to be more realistic....well most of the time:)

"Sometimes random concindences happen for no reason that anyone but a 19th century novelist would believe." (literature form the period is often convoluted and full of "miraculous" occurrences that would be improbable in reality).