Will the cat eat the mouse in your picture? :P

It = subject; in this case it is a preparatory subject, preparing for the long clause at the end. is established = predicate well = adverb; modifying the predicate that the mother's language does have an impact = clause corresponding to the preparatory subject "it". that = conjunction the mother's language = subject of clause does have = verb; emphasised affirmation an impact = object of clause.

That the mother's language does have an impact is well established,语序换成这样你可能好理解点，典型的主谓宾结构。但英语也有英语的语序，为避免头重脚轻，所以就换成了It's well established that the mother's language does have an impact.