Disclaimer: this is not an answer, and I am not an expert in the subject. I hope that the pointers I can give below are helpful nonetheless.

Some idea on motivic aspects of $t$-motives can be found on the website for a Banff workshop from 2009, see here.

There are several papers (including Anderson's original paper or this paper of Bornhofen and Hartl) which say that Anderson's pure $t$-motives are function field analogues of abelian varieties.
In this direction, I would think that the category of Anderson's pure $t$-motives is a function field analogue of the category of Chow motives of abelian varieties; similarly, the full category of Anderson's $t$-motives should correspond to the category of mixed motives generated by motives of abelian varieties (although this does not exist at the moment). Conjecturally (cf. this MO-question) the category of motives is generated by motives of abelian varieties. [Edit: This is expected *over finite fields*, thanks to Mikhail Bondarko and guest for pointing this out: if hom=num then the category of motives over the finite field is generated by abelian varieties and is the category described in Jannsen's "Motives, numerical equivalence and semi-simplicity".] If this is the right track, then Anderson's $t$-motives are the function field analogue of something that at the moment is expected to exist, but whose existence depends on serious conjectures...

In another direction, Anderson's $t$-motives seem to be analogues of motives for automorphic forms. Historically, they appeared after Carlitz modules were used to find function field analogues of cyclotomic extensions, and Drinfeld modules were used to find the function field analogues of the extensions of number fields arising from elliptic curves (see e.g. Rosen's book on number theory in function fields or Drinfeld's paper on elliptic modules). So one use of $t$-motives is the construction of Galois extensions of $\mathbb{F}_q(T)$. Furthermore, there are papers talking about uniformization of $t$-motives, analogues to lattices etc. which also supports the belief that $t$-motives should be function field analogues of motives associated to summands of locally symmetric varieties $\Gamma\backslash G/K$ (and hence to the motives that appear in the Langlands program).

I guess the above two directions can actually be explained in a bigger picture, but you'll have to ask an expert in $t$-motives or Langlands program about this...