This is a very, very broad question and there are no definitive answers. You should have a look at Mac Low (2013) for a review of the issues.
Star formation increased over the first few billion years of cosmc time and has been decreasing since (see your previous question). Most star formation occurs within galaxies and so it is the detail of the time dependence of the formation of galaxies and the time dependence of star formation within them that sets the overall time-dependence of star formation in the universe.
In basic terms, star formation occurs when you can get together large quantities of cold, gas and dust. In the local universe, the well-known Schmidt-Kennicutt laws relate the rate of star formation to the surface density of gas (to the power of something like 1.4) and show that star formation increases the more gas you can accumulate together.
Counteracting this are feedback processes (e.g supernovae, jets from supermassive black holes) which tend to heat and disperse gas. So from a qualitative point of view you can perhaps see why the star formation rate should peak. There is an initial period where baryonic mass is concentrated (helped by dark matter), star formtion begins, the pollution of the interstellar medium with metals makes it easier for gas to cool and form stars, but then once vigorous star formation gets going there are feedback processes which will act to throttle star formation. In addition, the expansion of the universe means that at later times there will be less merger-driven activity and gas that is accreted onto galaxies will tend to become less dense and hotter (because less dense gases are harder to cool).
I will stop there with my very brief answer (not least because I will be quickly out of my depth); these are all wide-open topics in astrophysics. Whilst the observation that the star formation rate of the universe peaked at $z \sim 2-3$ is pretty much settled, the explanations for why and when are not.
