Nadav answers this well, but since you want an explicit discussion of the entropy I will try to do that. Right after recombination, when the universe was a nearly homogeneous gas of hydrogen, helium, and photons, it was sitting at the top of three potential well: gravity, nuclear, and, dark energy or the cosmological constant. Thus it was at low entropy compared to what was possible if it fell down these potentials.
At first there was simple density perturbation growth by gravity which increases entropy by the fact that there are many ways this random process can occur. Then collapse continued via gas dissipation as gravity compressed and heated the gas which cooled by emitting many photons that roam the universe. The gas and dust collapse until stars form. This all also increases entropy by having many more random discrete objects.
Gravity compresses the stars which heat and begin nuclear burning at their cores so the nuclei start falling down their nuclear potential wells which was not possible in the earlier configurations. This generates gamma rays, more point particles, which convert to many more low energy photons by the time they escape from the surface of the stars.
Now we are seeing that space is accelerating its expansion so that increases entropy as this potential energy is transformed into more space.