"Science is an integral part of culture. It's not this foreign thing, done by an arcane priesthood."  Stephan Jay Gould

Why? A good question.  The answer at CERN's public portal

We can justify the LHC's existence on a number of levels:

Fundamental research

Update March 2019 because this page was referenced by Sabine Hossenfelder (for whom we have a lot of respect!): the following list was put up a long, long time ago. Things have changed. The Higgs boson has been discovered, LHC results have, so far, showed no sign of Supersymmetry etc... interesting times. Please note the "might" below.

This is CERN's core business. With the LHC the aim is to continue to push our understanding of the fundamental structure of the universe. The results from the LHC might shed light on:

• Dark energy
• Dark matter
• Extra dimensions
• Higgs
• Supersymmetry

For a basic introduction see the Atlas physics tour

Education

A huge number of technical students, Ph.D.s, post-docs, associates etc. pass through CERN

Spin-offs

• The World Wide Web for example

Knowledge transfer

• E.g. Detectors / Cameras for High Resolution Biology and medical use. 

  For example, the same method used in the CMS electro-calorimeter to study high energy physics and to search for strange new particles is used for medical imaging in positron emission tomography (PET) to study body functions and search for cancer.

Investment

• A lot CERN's budget is spent on hardware produced by European industry

International Relations

• A truly remarkable international collaborative effort