How does appendicitis infection cause a liver abscess?

Blood. The liver is the first line of defense for infection draining from the intestines including the appendix. For this reason, several types of infection in the intestine can cause liver abscesses, the two most common are appendicitis and diverticulitis.
Bacterial spread. Bacteria can enter the blood stream from an infected focus like appendicitis and progress upwards to the liver where an abscess can form.