The CMB originates from when the Universe became transparent, around 380,000 years after the Big Bang, and stars were born around 100 million years later. I know the first stars and galaxies aren't observable but I don't understand why this is the case if the CMB is observable...
The short answer is that there are many, many more CMB photons than there are from the first stars. When the CMB was emitted, the entire Universe was like the outer atmosphere of a Red Dwarf star with a temperature of around 3000 K. The CMB still completely dominates the electromagnetic radiation in the Universe.
So imagine the entire night sky radiating at the same intensity as a cool star (but, fortunately, redshifted into very cold microwaves). In comparison, the fraction of the sky covered by the first stars, the so-called Pop. III stars, is vanishingly small, and even worse, they are only a very small fraction of the photons we see in that wavelength range, so finding and identifying them is difficult.
EDIT: Also, we should possibly clarify something in the original post: We have not observed any of the first stars yet (as others have mentionded, there are candidates, but no really convincing detections yet). That does not mean they are not observable -- that would mean they were theoretically impossible to observe. We have no reason to believe it's not possible to observe them - we just haven't figured out how, yet, although we are probably quite close.
The key idea of the CMB is that it is constant throughout our entire universe. Essentially, because the big bang happened everywhere, and thus the CMB is seen from every point in the universe.
We detect the CMB today from parts of the universe which were ~13.8 billion light years away from where they were at the time of recombination (the moment visible, free light came into existence). So, the CMB is actually infrared blackbody radiation emitted when the universe was around ~3000K. Because of red shift from universal expansion, this radiation is now observed as microwave radiation.
Comparing this microwave radiation, constant in every direction, to seeing the earliest stars and galaxies - which are proposed to still exist - the light from them is difficult to capture. But there are papers that suggest we have found at least some stars and galaxies from the very early universe. I'll leave this paper as an example. https://arxiv.org/pdf/astro-ph/0703414v1.pdf