We suppose that $K$ is a category defined by a "species of structures with morphisms" $\Sigma$, the objects of $K$ are thus the sets endowed with structure of species $\Sigma$ and the morphisms those of $\Sigma$.
This is the language of Bourbaki. "Species of structures" are Bourbaki's version of categories. The analogue of an object is for example a set with some "structure", like a topology, or a group operation. Morphisms are functions that "preserve the structure". Rigorous definitions of these things and their morphisms, in full Bourbaki generality, can be found in Chapter IV: Structures of Theory of Sets.
Here first Grothendieck defines a solution to a universal problem in a category $K$. He's actually defining a limit, but for diagrams that look like a cover. He also defines a sheaf with values in an arbitrary category $K$, giving the extra condition on a presheaf. Then he restricts to the case where $K$ is made of objects with extra structures as above, and assumes that $K$ satisfies a further property: that a solution to a universal problem in $K$ remains a solution in the category of sets if we forget the structures. Then he says that in this case a sheaf with values in $K$ is also a sheaf of sets.