I am having a trouble in understanding the Example 4.7 (pages 65-66) in "Surfaces fibrées en courbes de genre deux", Lecture Notes in Mathematics, 1137. My French is not good and the Google translator confused me even more. Can someone outline this construction?

Especially, I don't understand how the branch divisor $D$ is defined on page 66. Also, does it follow from the construction that the fibration admits $-1$ sphere section? Thanks in advance.