Apparently the ﬁrst mathematical publication of Grothendieck where he uses the terms “functor” and “category” in the technical sense is the Kansas report.
From where Grothendieck had knowledge of categorial language?
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Grothendieck's familiarity with the categories predates Kansas. In 1948-1949 he attended Séminaire Cartan at École Normale Supérieure, where he "took the liberty of speaking to Cartan, as if to his equal" (Cerf's obituary). That would be Henri Cartan, the agebraic topologist, he is the son of Eli Cartan known for his contributions to Lie group theory and differential geometry (e.g. the differential forms). The theme of the year was simplicial algebraic topology and sheaf theory, developed by Leray since 1945. Grothendieck also attended a course on locally convex spaces, given by Leray himself at the Collège de France, so he probably heard the buzzwords. However, at the time Grothendieck felt out of his depth among people playing with the new algebraic toys "like they have known them from the cradle" (Récoltes et Semailles). In 1949-1953 he went to study at the University of Nancy under Delsarte, Dieudonné, and Schwartz among others. As Dieudonné wrote in Grothendieck's early work (1950–1960):
"When, in 1953, it was time to grant him a doctor's degree, it was necessary to choose from among six papers he had written, any one of which was at the level of a good dissertation. Of course, his fundamental paper on tensor products and nuclear spaces is the one that was chosen and it rapidly became a landmark in functional analysis."
According to Effros, this paper, Produits Tensoriels Topologiques et Espaces Nucléaires, already employed categorical methods:
"He was arguably the first to realize that the algebraic/categorical methods that flourished after the Second World War could be used in this highly analytic branch of functional analysis."
For 1953-54 Schwartz arranged for his brilliant pupil (who refused to become a French citizen because it entailed military service) a position in Brazil, and then Grothendieck spent a year (1955) at the University of Kansas, probably at the invitation of Aronszajn. It is there that his interests shifted from functional analysis to homological algebra. According to Malgrange, Grothendieck told him concerning topological vector spaces:"There is nothing more to do, the subject is dead."
And it was there, in Lawrence, Kansas, that Grothendieck wrote the "Tôhoku paper" Sur Quelques Points D’algèbre Homologique (published in 1956), extending the work of Cartan and Eilenberg on modules, where the categories appear in their birthplace environment. He also began to correspond with Serre, whom he knew from Paris and Nancy and who pulled him towards algebraic geometry, where Grothendieck will employ sheaf-theoretic and categorical methods to make his principal mark. The Grothendieck-Serre correspondence is translated into English. As Serre related much later,
"His knowledge of classical algebraic geometry was practically zero [...] I tried to help him with that. But... there were so many open questions that it didn’t matter."
Grothendieck confirms in Récoltes et Semailles that most of what he learned in geometry he learned from Serre, or taught himself, and calls Serre the "detonator" that provided the spark to ignite his explosion of ideas. In particular, it was Serre who described to Grothendieck the famous Weil conjectures in the cohomological language, a categorical bridge from homological algebra to algebraic geometry.
I found Jackson's As If Summoned from the Void biography sketch that tracks the origins of Grothendieck's ideas very helpful.