How do they produce styrofoam?

Answer 1

From Polystyrene beads.

Invented by Dow more than 50 years ago and identified worldwide by the distinctive Blue** color, STYROFOAM* products are the most widely recognized brand in insulation today. In the early 1900s, The Dow Chemical Company invented a process for extruding polystyrene to achieve a closed cell foam that resists moisture. Recognizing its superior insulating properties, buoyancy and "unsinkability," it was originally adopted in 1942 by the Coast Guard for use in a six-man life raft. That was the start of many other wartime applications by the Coast Guard and Navy.

Today, the Dow STYROFOAM brand includes a variety of building materials (including insulated sheathing and housewrap), pipe insulation and floral and craft products. But there isn’t a coffee cup, cooler or packaging material in the world made from STYROFOAM.

These common disposable items are typically white in color and are made of expanded polystyrene beads. They do not provide the insulating value, compressive strength or moisture resistance properties of STYROFOAM products. In order to protect the Dow trademarked name “STYROFOAM”, such other material should be referred to by the generic term “foam.”

Check sources for further information.

Sources:
http://www.dow.com/styrofoam/what.htm
http://inventors.about.com/library/inventors/blpolystyrene.htm
Polystyrene: http://en.wikipedia.org/wiki/Polystyrene