GARCH: The model is used for variance forecasting. For this reason it is a good tool for working with volatility. The model is generally given as:

at = etσt

GARCH is mean reverting. See pg. 29 (Lecture 6). The variance of the series is used to model a distribution with conditional mean = 0 and the conditional distribution is normal. See. pg. 32 (Lecture 6). The note by Engle is also a good introductory material to GARCH. Since you are modeling the error, the band or range would not be too big and since the movement of the error does not fall far from its own mean (mean of the error term), the generally picture you see on pg. 29 (Lecture 6) is a near linear. Here we are not predicting the estimated function that suppose to forecast the FOREX, but the distribution of the error of the FOREX forecast.

"Innovation" refers to the variance of the current error term in ARCH. In the previous term, there is a variance of the error. In this term, the error also change, may be due to volatility, this changes of variance in this term is called innovation. When employing GARCH, is the innovation get corrected? No, GARCH is not an error correction mechanism (ECM); it is a model use for predicting error of the "error changes" in the main forecast function. ECM is a separate matter.

Try to use STATA, it is best for now for ARCH - GARCH models, also Ox is good

Walter Enders book has a good practical approach. It is worthwhile trying to replicate its models. 

Regarding Matlab, It does support several garch variant, but it does not support regressor for variance model, it just accept regressor for mean model.