Although Putnam may never have really held the position he calls “metaphysical realism,” he was thought by many (including himself—see p. 489) to have held it, and this paper (along with others he wrote around the same time) was seen as marking a fundamental shift in his thinking. (The term “metaphysical” may recall the logical positivists’ concerns about “metaphysics.” There is a connection, but a better comparison is between “metaphysical realism,” as Putnam sees it, and the “transcendental realism” that Kant contrasted with his own “transcendental idealism.”)
The heart of Putnam’s argument against realism appears very early on (pp. 485-6), where he speaks of alternative satisfaction relations. (Remember that satisfaction is the generalization of truth as something like ‘true of’ that is needed to handle generalization and other “quantificational” sentences, so at the cost of a little less accuracy, Putnam might have spoken of alternative truth predicates.) His argument, which reappears in various forms for the rest of the paper, is that there is nothing available to distinguish which of these satisfaction relations is the “intended” one, the one that corresponds to what we really mean.
Putnam’s discussion becomes more explicitly Kantian in the latter half of the paper when he discusses the implications of these ideas for “internal realism” (the analogue of the “empirical realism” that Kant held). The clearest ties to the discussion on pp. 485f may be on pp. 494f, when Putnam discusses the sentence “‘Cow’ refers to cows.” (Notice that this is a sentence within the theory rather than speaking of its relation to an independent world, so it is something to be interpreted rather than something that expresses an interpretation of the language.)
Finally, comments on two minor points:
• Note that T1 on p. 484, which is a term, is something different from T1 on p. 485, which is a theory. Much of the time thereafter T1 will be a theory, but the variable will continue to be used for terms in diagrams later in the paper.
• Although Putnam’s reference to a “direct (and mysterious) grasp of Forms” (p. 487) may suggest (and may be) an allusion to Plato, you should think of Russell. Since universals are the only things outside our minds that we are acquainted with according to him, any tie between our beliefs and the external world must come by way of them, and acquaintance is reasonably described as a “direct (and mysterious) grasp” of universals.