[cbracken]

The process that caused the light we see as the CMB happened everywhere in the universe roughly all at once. At the time, from any given point, you'd just have seen blinding light but as things cooled, you'd see a sphere expanding around you where this light was dimming, as less and less, and finally none was being emitted around you, but due to the finite speed of light, light from outside that sphere would still continue arriving in your eyeballs. That's the situation we're in today; that sphere's just really big now.

During recombination epoch ~400,000 years after the Big Bang, this light would have been visible. Due to expansion, over time that light has stretched to longer and longer wavelengths, and we currently see it as microwaves.

Note: It's been ~30 years since I was actually studying physics & astronomy; others may be able to offer better explanations or correct me.