Could someone please point me towards a proof of why the image of a Galois representation on the Tatemodule of an abelian variety is limited by its MumfordTate group?

This is an immediate consequence of Theorem 2.11 and Proposition 2.9 in Deligne's 'Hodge cycles on abelian varieties' (notes by Milne available here: http://www.jmilne.org/math/Books/DMOS.pdf 2.11 shows that every Hodge cycle on an abelian variety $A$ over a field $k$ embeddable in $\mathbb{C}$ is absolutely Hodge, and 2.9 shows that all absolutely Hodge cycles are defined over a finite extension of $k$. So it follows that all Hodge cycles on an abelian variety have canonical $l$adic realizations that are defined over the same finite extension of $k$. In particular, an open subgroup of the Galois group fixes all Hodge cycles; that is, it maps into the MumfordTate group of $A$. 

