How do we know background radiation does not come from undiscovered galaxies? Each year while observing voids, astronomers discover more and more unexpected galaxies within them.  


How do we know the cosmic microwave background radiation is not the weak light from distant galaxies? 
How do we know cosmic microwave background radiation is not stars' light reflected by dark objects?
Assuming you mean the Cosmic Microwave Background ...

The CMB is entirely in a very narrow range of the microwave part of the spectrum. Galaxy emissions are all over the spectrum.

The CMB comes uniformly from all over the sky. It's like looking at painting that is all one color. Galaxy light looks like pinpoints on a dark background. (Nearby galaxies are not pinpoints, but those at cosmological distances look that way.)

The CMB matches the exact wavelength we expect for light that is heavily redshifted from the moment of last scattering. That moment was hundreds of millions of years before galaxies ever formed. Light from oldest galaxies are at a redshifted wavelength from 500 million years after the last scattering.

If the CMB was due to early starlight reflecting off dark objects, you would see a very different spectrum. It would contain many absorption lines as the dark objects absorbed some wavelengths better than others. Also the reflected starlight would be across a wide range of frequencies, unlike the CMB which has a very narrow range.