# what does the constant term in the Moving average model represents?

that equation is gotten from here.

Is that mean term represents the best fit for the bias term for MA model gotten by minimizing the mean squared error equation?

It depends what you mean by mean square error equation. No, $\mu$ is not the "best fit" parameter estimate; it is just the unknown parameter. It is some unknown number that could be estimated, but right now, isn't. Once you estimate it based on data, getting a specific number, it will be the "best fit." Or are you talking about predicting future data assuming the parameters are known? In this case, your "mean square error equation" could be referring to the forecast error. If this is the case, then you will need to specify if your forecasts condition on any data being known, and how far back this conditioning will go (infinite past or finite past).
This parameter, whatever specific number it is, does represent the unconditional mean of the process as $E[X_t] = \mu$, by linearity of expectations.