TITLE: Why higher category theory?
QUESTION [85 upvotes]: This is a soft question. 
I am an undergrad and is currently seriously considering the field of math I am going into in grad school. (perhaps a little bit late, but it's better late then never.) I have some background in commutative algebra and algebraic topology and I am learning AG quite intensively. I understand that category theory is useful for generalizing different mathetical objects, and in the field of AT, it makes the idea of homology and homotopy so clear as they are just functors from (based) space to the category of (abelian) groups, and many things are just limits or colimts; I also do understand you need the notion of abelian categories for homological algebra and  constructing sheaves like the grand project done by Grothendieck.
However, I still don't understand why study such elusive things like higher category theory where many things defy intuition? I have talked to several top mathematicians in my department, many of whom apperently think it is useless. However, it should not be as celebrated mathematicians like Jacob Lurie is making a lot of effort trying to understand it. So can anyone explain to me why study higher category theory? What problems are they most concerned with in higher category theory and what connections are there with AT or AG?
Thank you very much in advance.

REPLY [3 votes]: I refer you to a talk I gave in Paris last June on "The intuitions of cubical sets for nonabelian algebraic topology" which explains how from my 1967  groupoid version of the Seifert-van Kampen Theorem I was led to ask about  the potential relevance of groupoids in higher homotopy theory. Eventually with Philip Higgins we obtained a new formulation of basic homology/homotopy theory using actual compositions of cubes, and avoiding Poincar'e's "formal sums", now expressed in terms of free abelian groups.  The details are in the EMS Tract vol 15  (2011) advertised here, where there is a pdf available. This book also discusses the history and intuitions.
These methods give more understanding and also homotopical computations not available by other methods. As an indication, I mention the bibliography on the nonabelian tensor product which has 131 items, mainly in group theory. Relatively few algebraic topologists have worked in this area. Here is a short exposition related to the Blakers-Massey Theorem.
These papers use higher groupoids as concrete algebraic objects better able to express higher geometric ideas than purely $1$-dimensional constructions.
What is called "higher category theory" has not yet, it seems to me, accommodated the cubical approach. However for the work in which I have been involved the cubical methods have proved essential.
See also this mathoverflow answer on higher homotopy groupoids.