What is best way to compact styrofoam and discard it?
I'll assume from your question that you live in a place where it's not recyclable.

Here's an idea: turn it into a resource and use it to improve your house's insulation! I am currently re-using strips of styrofoam packaging material to insulate around my new windows. You could put the bigger sheets of it on your attic floor and then cover it up with cellulose. Etc.

You could offer it up free on Craigslist too. You'd be surprised what people can do with what you might consider a waste product.