The radiation was "produced" about 380,000 years after the Big Bang, and it was produced at every point of the Universe. From the very beginning, it's been (almost) uniform (the same at all places) and isotropic (the same in all directions). Since that time, the radiation was moving in all directions, essentially without any interactions.
The cosmic microwave radiation "decoupled" - separated - from the rest of the matter in the Universe in this era we call "decoupling". Before the decoupling, the temperature of the Universe was so high that electrons and protons were largely separated in a plasma filling the Universe. Plasma carries a lot of random electric charge that severely interacts with photons all the time - so the plasma was opaque for the radiation.
However, after the "decoupling", the atoms were formed for the first time - mostly Hydrogen atoms. Hydrogen atoms are neutral and their interactions with the photons are much weaker so the Universe became essentially transparent. The photons - and everything else in the Universe - at the moment of decoupling had a certain high temperature (thermal equilibrium, about 3000 Kelvin) which means that their spectrum was Planck's black-body thermal spectrum corresponding to the temperature.
From that moment, photons were moving without any interactions and their wavelength was increasing proportionally to the size of the Universe. That also means that the energy of each photon was decreasing by the same factor; the temperature of the black body radiation did the same thing. That's why the current CMB temperature is just 2.7 Kelvin. You may see that the Universe's linear dimensions expanded about 1,000 times from the decoupling.
(Note that 13.73 billion years over 380,000 years is substantially more than 1,000. That's because in the early stages, the expansion of the Universe was "decelerating" as a function of time. Only in recent few billions of years, the expansion got actually accelerating because of dark energy that gradually became important.)
When the WMAP probe detects a photon of the cosmic microwave background, this collision with the telescope is the first interaction of this photon since the moment when the Universe was 380,000 years old. This fact allows you to to deduce how far is the point when the photon was born - or when it last interacted with another object. The birth place is clearly a point in the direction where the photon is coming from. The distance is always the same so all the photons we see here today had to be produced at a particular spherical shell in spacetime. The center of the shell is "our place in the past" and the radius is such that the photons from the shell, when travelling inwards, exactly needed those 13.7 billion years of the cosmic time to get here.
Individual photons certainly don't have a rest frame. However, there is a rest frame in which the CMB is almost perfectly isotropic (the deviations from a perfect blackbody spectrum are of the order of 1 part in 100,000), and for convenience we call that the rest frame of the CMB.
That frame is essentially the rest frame of the plasma which emitted the CMB, i.e. the surface of last scattering, adjusted for the Hubble flow.
Our motion causes anisotropy through simple Doppler shifting: the CMB photons coming from the direction we're currently heading towards get blueshifted, the photons in the opposite direction get redshifted.
The Earth's velocity with respect to that frame is a little complicated, because we're orbiting the Sun, which is orbiting within the galaxy, which has its own motion in the local group, etc. Of course all of those motions are operating at different time scales, and different speeds. The shortest period effect is of course due to our orbit around the Sun, but our orbit speed is pretty sedate compared to the other motions I mentioned. So there's noticeable annual variation in the exact amount and location of the anisotropy, but the long period high velocity motions are the major factors controlling the anisotropy.
This famous image (from Wikipedia)
shows the CMB from WMAP after the dipole anisotropy has been subtracted. The 1 in 100,000 parts variations I mentioned above are amplified enormously, otherwise the image would look totally uniform. This amplification can only be done after the anisotropy compensation, otherwise the anisotropy would totally dominate the image.
Here's the dipole map from the COBE data, courtesy of NASA:
blue corresponds to 2.721 Kelvin and red is 2.729 Kelvin.
Best Answer
I found this answer at Professor Douglas Scott's FAQ page. He researches CMB and cosmology at the University of British Columbia.
“Where does it come from?” is also answered:
EDIT:
@starwed points out in the comments that there may be some confusion as to whether someone in the rest frame is stationary with respect to the photons in the rest frame. I found a couple more questions on Professor Scott's excellent email FAQ page to clarify the concept.