The sheer amount of discussion, confusion, and controversy here is remarkable. I think this goes back to a fundamental issue: the question of "How?" or "Why?" something happens leads to an infinite regress of "How" and "Why" questions: you can always ask more "Why" and "How" questions unless you eventually reach some concept which you can agree on, somewhat like an axiom.
Richard Feynman was once asked a question along the lines of "Explain the attraction between magnets." After hemming and hawing for several minutes, he eventually said something to the effect of "I can't explain the attraction between magnets because I don't understand it in terms of anything else I'm familiar with." The point is that there are some axiomatic things one must understand without regard to other more familiar objects, and it is these axiomatic things on which one bases their understanding of other objects.
In fact, I think that exact Feynman interview touches on a lot of the issues people bring up in this thread, such as how contact forces are actually microscopic electromagnetic forces, etc. Here is a link:
http://youtu.be/wMFPe-DwULM [skip abound 6 minutes if you want to avoid the hemming and hawing. But watching it shows that questions like this can be head-scratchers even for a Feynman.]
The question in the video has to do with the attraction between magnets, but I think it applies equally well to the idea of an electromagnetic field.
I think though that my initial answer--there are actually particles mediating the fields--is actually a deep answer, albeit nonclassical. Modern physics says that particles and fields are unified in the same entity (the quantum field), so one can view pretty much anything as both a particle and a field, so there's really no distinction between contact and noncontact forces. [So what DaleSpam said, "Particularly since in modern QFT matter is made of fields too." is just one way of looking at things--one might equally say that the field is made of particles.]