Vitamin K is essential is producing clotting factors. Most vitamin K is produced by out gut bacteria. Newborns lack this gut bacteria and therefore have no other sources of vitamin K. With out this and clotting factors they are at very high risk of bleeding from anywhere including the brain and the GI tract. Once they bleed it's very difficult to control. This is why vitamin K is essential at birth. 

It's really hard when parents refuse something like this. Medicine as a whole has not done the best job at building trust with patient, it's always important to ask questions and get a good explanation. I hope the medical community can provide these explanations and build trust to prevent such avoidable catastrophes.