Mark Sapir's question inspired me to ask the question in the title. A lot of mathematicians who have done work related to mathematical physics or, in fact, are mathematical physicists themselves (Kontsevich, e.g Kontsevich, Stasheff, Getzler, Manin, etc.) have done work with operads, but I don't really grok why operads have anything to do with physics. I wonder if anyone has ideas about why there is such a close connection.
Added: Examples of how operads are used in physics are welcome, but just like in Mark Sapir's question, I am much more interested in a general reason that explains why they appear. Especially why multiple operads appear, not some particular operad. Note that operads can be defined in various categories (topological spaces, vector spaces, etc) and I'm most interested in operads in the category of vector spaces. But if there is an answer that covers more than one category, that would be very nice.