I don't believe you'll get a better than "smash rocks, see what happens" understanding of what the LHC does without significant time invested in understanding.
That's okay: politicians don't actually need to know how it works, they need to know why they should build it -- which is a) something they have domain expertise in and b) a much easier question.
And do you really expect politics to raise funds for such a project if nobody can explain to them what they are doing?