What is so special about the way Moving average models, Autoregressive models and their combinations (ARMA, ARIMA) are defined that they seem to fit many of the univariate time series we observe in our world ? What is so unique about the way many dynamical systems work that these models or a combination of these model seem to fit them ?
"they seem to fit many of the univariate time series" is a bit of an exaggeration. Over a decade of performing various applied data analyses, I have noticed that more often than not a stochastic volatility component is necessary. However, if you focus on calculating conditional means only and regime switches are infrequent, ARIMA may serve as a decent approximation.
Why does ARIMA work? As one person told me: "If you are here today, the chances are you will be here tomorrow". The world does not change overnight. So, typically, the values today are important for predicting the values tomorrow. Shocks today are important for predicting shocks tomorrow. Hence the AR and MA components.
Why do we need differencing in ARIMA (integration of order d) instead of just sticking to ARMA? Because most real-life processes are not stationary. They are going somewhere. Differencing the data d times may solve this in some (but not all) situations.
Why is ARIMA a viable research tool? Because, unlike very complicated models, it boasts estimation methods which are relatively simple, statistically efficient and robust. So quite quickly the researcher can get an answer which has informative value (even if some other method would provide more information).