We can see the initial density fluctuations as temperature fluctuations in the cosmic microwave background (CMB). Almost all of the CMB was causally disconnected at its emission time, as the horizon scale at the time is around 1 degree on the sky. We see temperature variations larger than that, and since they are not causally connected, we know that they must have been frozen in time since whatever process created them in the much earlier universe. (Likely inflation, as I noted in another comment.)

Also, gravity can only amplify already existing density variations. Thus the smaller-scale (causally connected) CMB temperature variations, and the density variations in the universe today (responsible for galaxies and larger-scale structure), must have originated from similar initial density variations. In fact we understand quite well (mathematically) how density variations gravitationally amplify over time, and a wide range of observations generally all point to initial density variations having essentially the same average amplitude at every scale (the one part in 10-100 thousand that I mentioned).