Why Moving Average model does not contain past time series values?
That is due to the definition of the moving average (MA) model. The dependent variable $x$ is supposed to be a linear combination of the current and lagged innovations (shocks) $\varepsilon$. That is how the model is defined.
It may or may not be a reasonable model for the particular data you have -- but that's another question. At least you have a choice to try the MA model out.
There is an intuition to the MA model, too. If a process $x$ reacts to a shock $\varepsilon$ with a delay, then a lagged shock is a natural variable to have in the model -- and that happens to be the case with the MA model.
Regarding estimation, the MA model can be fit using maximum likelihood. You specify a likelihood function and then use an optimization routine to obtain the likelihood-maximizing parameters. MA model can be represented in terms of a state space model, and Kalman filter comes into play there -- if that helps.
Regarding the lagged values of the dependent variable, you may look at the autoregressive (AR) model or the autoregressive moving-average (ARMA) model instead. There you have lagged values of the dependent variable on the right hand side. However, note that a MA model has an AR representation, under some conditions. Thus essentially the difference between the AR and the MA models is not that large. It is about looking at the same data generating process from a different angle and representing the same object in different ways.