Question

why appendix is the vestigial organ in human digestive system

Answer 1

The vermiform appendix still serves a purpose. It is not an indispensible organ but it does something. The reason why it's actually there isnot 100% sure but it does make sense. What we know is that it is full of limphoyd tissue. This means that it has an important role for the maturation of lymphocytes (a type of white cells of central importance in the immune response). A very large number of lymphocytes are 'stored' in the appendix. The reason why it is there is to 'filter' the content of the intestine before it goes from the small intestine to the large intestine (the appendix is at the border) and keep the bacterial population under control. The gastrointestinal tract is a main way of acces for microrganisms and is therefore necessary to have a large base for immune cells to stay near it. 
Lymphoid tissue is also all along the intestine. This is why you can remove the appendix without any particular consequence.

The tonsils have the same function. If they are removed, other surrounding lymphoid tissue can grow and compensate this loss. Nevertheless they are important as they are at the entrance of the gastrointestinal tract and can block microrganisms before they pass into the pharynx or larynx.

MARK BRAIN...

Answer 2

The human appendix had been proposed to be a vestigial structure, a structure that has lost all or most of its original function or that has evolved to take on a new function, because it is thought to be the shrunken remnant of the cecum thought to have been present in a remote ancestor of humans.