I haven't been able to find a reference to the first who showed that the small density fluctuations in the cosmic microwave background led to the formation of large scale structure as we know it.
I'm not an expert in this, but I'll give it a try.
I think you've got the argument the other way round here. This was never discovered. It was assumed. But why?
- We know for the CMB being a redshifted blackbody-spectrum. This means the CMB consists of photons that were emitted in interaction with matter.
- The CMB fluctuations inform us about the depths of gravity wells that those photons had to escape from (amongst other things). Thus we get a picture of the baryonic and dark matter density at the time of photon-matter decoupling.
- This allows us to set up linear perturbation theory + simulations to look what happens to a uniform gas with the measured fluctuations on top. That's the rough idea.
You could now of course ask "who was the first to implement this?". The earliest theoretical work that I'm aware of (Peebles 1965) linking "a possible universal blackbody radiation" with the formation of galaxies came in the year of the discovery of background radiation by Penzias & Wilson 1965.