Stars have a temperature of 3000 - 30,000 K. You would think that the billions of stars in the sky would completely mask the paltry 2.7 K background radiation, and distort its temperature. But in fact, it is not the temperature of objects that is 'additive' but the corresponding intensity or brightness of the light that the objects produce which is additive.
Max Planck showed that radiation that is in thermal equilibrium with matter follows a simple intensity curve when you plot the brightness of the radiation against the frequency at which the radiation is measured. This curve depends on only one parameter; the temperature of the body. Stars at a temperature of 6000 K emit most of their light in the visible region of the electromagnetic spectrum, and in particular the peak of this curve is near the 'yellow' part of the visible spectrum, at a wavelength determined, in microns, by 2897/T or (2897/6000) = 0.48 microns. Radiation at a temperature of 2.7 K, on the other hand, peaks near 2897/2.7 = 1079 microns or 1.079 millimeters, which is in the microwave part of the spectrum.
The second feature of black body spectra is that the intensity at any wavelength can be exactly computed once you know how luminous the black bodies are. The cosmic black body radiation contributes about 10 billion photons for every proton and neutron in the universe. Stars in the Milky Way, and for than matter in all the other 50 - 100 billion other galaxies in our visible universe contribute only about a few million or so ( my rough guess). When you calculate how bright the stellar light is at a wavelength of 1 millimeter where the cosmic background is brightest, the Planck black body curve indicates that the brightness of the stellar light is about ( .28/1079)^2 or 6.7 x 10^-8 as bright as the stellar light was at optical wavelengths. When you further add in the fact that stellar photons are outnumbered by about 10,000 to 1 compared to the cosmic background light, you find that at microwave wavelengths, starlight has virtually no DIRECT contribution. It can, however, contribute by a chain of indirect steps.
Young stars are surrounded by dust, and massive young stars can ionize the gas in their surroundings. This means that if a galaxy is currently producing lots of young massive stars, it can be a strong source of infrared and radio radiation with a brightness that equals or exceeds the combined luminosity of all of its visible starlight. Supernovae remnants produce yet another kind of radio emission ( synchrotron emission), as do the electrons trapped in the magnetic field of the Milky Way. This infrared and radio radiation can interfere with the detection of the cosmic background light. For our Milky Way, the radio emission from its young star clouds and synchrotron electrons is as bright, or brighter than the cosmic background light at microwave wavelengths, and has to me masked-out in order to detect the cosmic background light. Also, it is hoped that with sensitive enough radio telescopes we will able to detect this same galaxian light signal in very distant galaxies as a 'mottling' of the cosmic background light.