Why is the homogeneity problem actually a problem? I don't mean to be flippant but I have never understood why the homogeneity in the universe needs.explaining. I have searched other questions on stack exchange and they don't really address the question. I am going to ask.
Say a bunch of identical particles are created contained in a void of absolutely nothing. They are endowed with a set of quantum laws and the quantum mechanics is deterministic. Initially, they are in thermodynamic equilibrium. They begin to expand so fast that they break into regions that are causally disconnected. Then each of these regions is identical, has the same set of laws and zero external influence by definition. Surely they are each essentially an initial. value problem with the same start conditions. Mathematically they should all evolve the same, shouldn't they? Even if QM were not a deterministic theory, these regions are identical, isolated and have the same laws. How could they evolve differently? And, unless they either have different sets of laws, different starting conditions, or some differing external influence from another source, how could they ever be different?
 A: As you mentioned, the quantum fluctuations that occurred in the early universe grew via a process called Jeans Instability and formed the cosmic structure that we observe.
As you know there are four components in the universe. These are dark energy ($\varepsilon_{\Lambda}$), dark matter ($\varepsilon_{DM}$), baryonic matter ($\varepsilon_{b}$) and radiation ($\varepsilon_{r}$).
First of all, $\Lambda$ does not fluctuate since the energy density is constant. ($\varepsilon_{\Lambda} = \varepsilon_{\Lambda,0}$)
During the RD(radiation dominated) universe CDM and BM fluctuations cannot grow   inside the horizon (due to Meszaros effect and baryon-photon decoupling). At the MD(matter-dominated) universe CDM starts to fluctuate however BM cannot, since its still coupled to the radiation. So CDM fluctuations start to grow. When the BM decouples from radiation (occurred around $z\approx 1000$), it falls into the potential created by the CDM particles.
Since the density fluctuations of these components evolve differently during the RD and MD universe, we cannot expect perfect homogeneity and isotropy. Also, currently, the universe is not perfectly homogeneous or isotropic. We are just making an approximation or assumption.
Since this is a complex topic, I cannot explain the details of it here. However, If you want to learn about how these fluctuations evolve I recommend you to read these books and articles.
1 - Cosmology: The Origin and Evolution of Cosmic Structure 1st Edition
by Prof Peter Coles, Francesco Lucchin
2 - Christos G Tsagas. Cosmological perturbations (https://arxiv.org/abs/astro-ph/0201405)
3 - David Tong. Structure formation, 2019. http://www.damtp.cam.ac.uk/user/tong/cosmo.html
4 - Malcolm S. Longair. Galaxy Formation (Astronomy and Astrophysics Library)
