"I'd like to know more" is the vaguest possible question :-) The best you can do is to watch the corresponding talk here - it is fantastic ! - and probably come back with more questions, since it is quite dense.
The background for the statement you are asking about is roughly the following (I am no expert and writing from memory - the following may contain nonsense and should not be taken literally):
Consider the moduli stacks $\overline{\mathcal{M}_{g,n}}$ of stable curves of genus g with n marked points. These are interconnected by a bunch of maps corresponding to glueing together stable curves, forgetting marked points, collapsing components to restore stability etc. (these are all operations with an intuitive geometric meaning and are wonderfully explained in Kock/Vainsencher's "Invitation to Quantum Cohomology").
If we map these into the category of motives, we can for each $n$ form their direct sum over $g$, thus getting a sequence of motives, and then assemble the above maps to form the structure maps of an operad, i.e. mainly "composition maps" $\overline{\mathcal{M}_{*,n_1}} \times \ldots \times \overline{\mathcal{M}_{*,n_k}} \rightarrow \overline{\mathcal{M}_{*,n_1 + \ldots + n_k}}$.
The word "motives" here means Chow motives and to make sense of this for stacks, one has to extend the definition of Chow groups to suitable stacks and then apply the usual constructions. This can be done in at least two different ways (Vistoli's and Toen's, see the talk for references).
Then the statement in the talk is that the motive of any smooth projective variety ("projective scheme" is a bit too bold because we are using Chow motives) is acted upon by this operad object in motives. To construct the action morphisms one uses cycles (and thus morphisms in motives) $I_{g,n}(V)$ corresponding to certain quantum cohomology classes, which are present in the quantum cohomology of any smooth projective variety...
Of course if you apply now any monoidal functor to all of this (e.g. a cohomology) you get an operad, and an object with an action from it, in the target category...