What is the biological reason for a burst appendix being potentially lethal?
Necessary conditions

In order for the process culminating in appendiceal rupture to begin, there must first be proximal obstruction of the lumen (the inside cavity of the appendix) that prevents normal communication with the bowel.


  Obstruction is most commonly caused by a fecalith, which results from accumulation and inspissation of fecal matter around vegetable fibers.
  The bowel wall inside the (now closed) appendix continues to secrete muscous, and bacteria continue to multiply, producing the distention that is often perceived as nausea, vomiting, and poorly localized abdominal pain.1 


The process leading to rupture  

As the pressure in the appendix increases, venous pressure is exceeded. This means blood can no longer exit the venules of the appendix normally. In the presence of ongoing arterial flow, this leads to vascular engorgement, worsening the distention and increasing pressure further. The inflammatory process spreads from the lumen (inside) of the appendix to the serosa (outside) and then to the parietal peritoneum (lining of the abdominal cavity). This last step is what causes the characteristic shift in pain from poorly localized visceral pain to focal right lower quadrant pain. 

As distention worsens, arterial pressure is eventually exceeded, and the most poorly vascularized areas of the appendix (generally just beyond the point of obstruction on the antimesenteric border) infarct first. Dead tissue has very little structural integrity, and once infarction happens perforation is likely to follow quickly.

Why this is (sometimes) lethal

There are several possible mechanisms, more than one of which is probably operative in most cases. 

The three I have listed below all converge on the same final common pathway: In the absence of adequate systemic blood pressure (shock), the heart is not able to pump normally and will eventually deteriorate into a non-perfusing rhythm (a.k.a death).


Bacteria in the peritoneal cavity:
As in bowel perforation of any cause (e.g. perforated peptic ulcer, perforated diverticulum, traumatic perforation), bacteria in the peritoneal cavity (the “free space” outside the bowel within the abdomen) triggers an intense inflammatory reaction and fluid sequestration. Fluid pulled from the circulation into the peritoneal space causes hypotension and (hypovolemic) shock.  
Infarcted bowel: Dead tissue connected to the body is never a good situation. Intracellular contents are released into the bloodstream – electrolyte release can cause hyperkalemia and other electrolyte imbalances; reactive oxygen species can lead to hypotension and (distributive) shock.
Sepsis: Bacteria (generally gram negative rods from among normal gut flora – most commonly Escherichia coli or Bacteroides fragilis)
may enter the bloodstream due to compromised vessels during the process of infarction and rupture. Lipopolysaccharides in the cell wall of gram-negative bacterial organisms lead to activation of monocytes and macrophages. This results in increased levels of interleukin-1 and tumor necrosis factor and eventually of IL-6 and IL-8. These cause widespread endothelial injury leading to diffuse alveolar damage in the lungs and activation of the coagulation system leading to disseminated intravascular coagulation. These cytokines also lead to systemic vasodilation and (distributive, septic) shock.





References


1. Silen W. Chapter 300. Acute Appendicitis and Peritonitis. In: Longo DL, Fauci AS, Kasper DL, Hauser SL, Jameson J, Loscalzo J. eds. Harrison's Principles of Internal Medicine, 18e. New York, NY: McGraw-Hill; 2012.


2. Kemp WL, Burns DK, Brown TG. Chapter 8. Hemodynamics. Pathology: The Big Picture. New York, NY: McGraw-Hill; 2008.


3. Charles Brunicardi, F. Acute Appendicitis. Schwartz's Manual of Surgery; 2006.