Thanks to Sanath for collecting relevant links to the nLab entries in his reply. I'll just add some more remarks.
First of all it seems relevant here that while "motivic mathematics" is a big field, and while those motivic multiple zeta values which are used in expressing scattering amplitudes do originate in this field, as far as I can see little to no actual "motivic mathematics" appears in most (or any) articles on scattering amplitudes, instead these articles simply use some simplifying relations provided by that theory.
To orient oneself here, it is important to notice the following deep relation (I had another PO message on this recently here):
what physicists call an analytically regularized trace of a Feynman propagator and hence a vacuum amplitude is mathematically a zeta function. Now the bulk of number theory is all about the observation that in some deep and slightly mysterious analogy (which is sometimes referred to as the extended function field analogy) one may construct zeta functions not just from "wave operators", for instance form the Laplacian on a Riemann surface (used in computing the vacuum amplitude for the string) but also from the data provided by rings called "number fields" and (and function fields). The big idea here is that it makes sense to regard such a number field, purely number-theoretic as it may seem, as the algebra of meromorphic functions on some arithmetic analog of a Riemann surface. It is in this analogy that number theory touches string theory. Accordingly the "ring of integers" of a number field is analogous the structure sheaf of a Riemann surface, and accordingly its ring-theoretic spectrum is a variety -- an "arithmetic variety" -- which is much analogous to a Riemann surface.
The point here is to see that number-theoretic zeta functions assigned to such arithmetic varieties have a reason to be related, somehow, to zeta functions as they appear in vacuum amplitudes in physics. And the same story goes through for more general amplitudes, which have expressions of the form that mathematicians call "multiple zeta functions", because they depend on more than one variable.
Studying scattering amplitudes in physics comes down to handling loads of such multiple zeta functions, and a technical problem is that these have an intricate combinatorics which often makes it hard to reason about them and deduce properties of scattering amplitudes.
But -- and here is finally where the motives come in -- in arithmetic geometry one found that there are zeta functions and the like ("motivic L-functions") assigned not just to varieties, but also to "their motives". Here "the motive" of a variety is really just some simplified incarnation of that variety, namely it is that variety regarded in a category where morphisms are somehow "linearized correspondences" between varieties. This itself has some deep relation to quantum theory (to the fact that quantization is a kind of linearization of non-linear geometric mechanics) but for just understanding what people in scattering theory have in mind when they mumble "motivic motivic motivic" this has no relevance.
Instead, all what matters is this: the mathematicians found some way to think of these motives of their varieties and so forth, and in the end they came up with new kinds of functions, called "motivic multiple zeta functions". The name refers to how these were obtained, but in the end they are just some functions of a bunch of arguments which are similar in structure to oridnary multiple zeta functions and hence to expressions as they appear in scattering amplitudes. But by the magic of motivic number theory etc. pp. one has some identities relating the ordinary multiple zeta functions to those other functions called "motivic multiple zeta functions".
And the upshot is simply that re-expressing scattering amplitudes in terms of these other functions, these "motivic zeta functions" (which in the end are just some functions of some variables) may make some combinatorics of scattering amplitudes considerably more tractable.
So what I am trying to say is that on a first level "motivic structures" in scattering amplitudes is something very mundane. It just refers to the fact that one may open a math text that has the word "motive" in the title, find in their some identity between some functions, observe that these are functions one happens to be looking at, too, and then using that identity in one's work.
On a second, deeper level, then one may of course wonder "why" it is that the theory of arithmetic motives that ends up lending a hand in simplifying expressions for scattering amplitudes. It seems to point to some deep mysterious relation. Most readers here will be happy to ignore this "why" question and any speculation it may lead to. But this is where Kontsevich's idea from 1990 comes in, that the there is naturally a ("cosmic") Galois group action on the space of deformation quantizations and that this controls the relation of scattering amplitudes to periods and arithmetic geometry. This is what you find pointed to at some of those links (e.g. here) that Sanath pointed to in his reply.