As commented in the episode, Kuhn doesn't explicitly relate his insights about the history of science to more familiar philosophical accounts of epistemology until his 1969 postscript. I thought I'd give some of the text he uses there so folks can read it for themselves. This is from page 192-193; you can see the context here.
If two people stand at the same place and gaze in the same direction, we must, under pain of solipsism, conclude that they receive closely similar stimuli. (If both could put their eyes at the same place, the stimuli would be identical.) But people do not see stimuli; our knowledge of them is highly theoretical and abstract. Instead they have sensations, and we are under no compulsion to suppose that the sensations of our two viewers are the same. (Sceptics might remember that color blindness was nowhere noticed until John Daltonâ€™s description of it in 1794.) On the contrary, much neural processing takes place between the receipt of a stimulus and the awareness of a sensation. Among the few things that we know about it with assurance are: that very different stimuli can produce the same sensations; that the same stimulus can produce very different sensations; and, finally, that the route from stimulus to sensation is in part conditioned by education. Individuals raised in different societies behave on some occasions as though they saw different things. If we were not tempted to identify stimuli one-to-one with sensations, we might recognize that they actually do so.
Notice now that two groups, the members of which have systematically different sensations on receipt of the same stimuli, do in some sense live in different worlds. We posit the existence of stimuli to explain our perceptions of the world, and we posit their immutability to avoid both individual and social solipsism. About neither posit have I the slightest reservation. But our world is populated in the first instance not by stimuli but by the objects of our sensations, and these need not be the same, individual to individual or group to group. To the extent, of course, that individuals belong to the same group and thus share education, language, experience, and culture, we have good reason to suppose that their sensations are the same. How else are we to understand the fulness of their communication and the communality of their behavioral responses to their environment? They must see things, process stimuli, in much the same ways. But where the differentiation and specialization of groups begins, we have no similar evidence for the immutability of sensation. Mere parochialism, I suspect, makes us suppose that the route from stimuli to sensation is the same for the members of all groups.
Wes quickly identified this as something like "evolutionary Kantianism," meaning that there is something out there (the Thing-In-Itself), but it gets processed by us in some way to make the objects of experience. For Kant, this is done merely by way of our species makeup, for Kuhn, paradigms are also involved. In both cases, "the world" is objective (contra Randians) in that we cannot change the world of our experience at will or whim. Kant is not a relativist, in that "the world," meaning the world of phenomena as investigated by science and about which we can have knowledge, does not change given different theoretical apparatus or cultural assumptions.
Is Kuhn a relativist? Well, it depends what "the world" refers to, and Kuhn's break-down of the elements in play between stimuli and sensations is, I think, a fruitful alternative to Kant's distinction between the Thing-in-Itself and the Phenomenal World. For Kant, science is about phenomena. Those do transcend any particular perception we have of them, any particular human experience (though not all human experiences); the Phenomenal World does provide a whole, rich, endless realm for investigation. For Kuhn, stimuli, i.e. whatever is really causing our sensations, are ultimately what we're shooting to describe in science.
What complicates this is that Kuhn opposes this not to sensations exactly, which are individual, but the "objects of our sensations." The distinction is not between "the given," i.e. sensations, which we then abstract from to create percepts, because Kuhn doesn't believe in a theory-free "given," in raw sense data of the sort advocated by British empiricists. Sensations themselves are colored by a paradigm, and sensation delivers to us not just colors and flavors but objects of perception, which are not causes of perception (it's the stimuli, together with many physiological and environmental factors, that cause the perception) but are given by perception. The objects of perception are not theoretical posits in the sense that the stimuli are, but they look the way they do in part because of our theories, and we could potentially (though this is not likely as a practical matter, as paradigm shifts don't just happen every day) change theories and see them differently.
You might say that the objects of perception are the stimuli as perceived through the filter of our physiology plus our paradigm (using the rose-colored glasses image often inaccurately used to characterize Kant's position). This would make the stimuli the logical limit of inquiry: all theorization attempts to uncover these more fully, with less distortion. Still, it's out of the question that we can have "the full truth," i.e. grasp the stimuli in their fullness, not because Being is ever hidden or some crap like that, but because a state of affairs is by its nature transcendent: it has infinite nuances, and there is always the possibility of new perspectives on it, new maps of the terrain, and a scientific theory is like one of those maps. To make the metaphor even more literal: do we ever see "the terrain itself" in its fullness? Not when we could always explore underground, or at the microscopic level, or as seen from space. To say that a stimulus causes a perception is an oversimplification, bringing to mind the picture of a single object with light waves bouncing off of it reaching our eyes, but really, it's the total situation that results in a particular sensation, not a single object, and what we count as an object is a result of our paradigm.
It's sounding here like Kuhn's stimuli really are like Kant's Thing-In-Itself: something fundamentally unknowable. But I think we can interpret Kuhn more charitably here, and interpret my difficulty in mapping Kuhn to a traditional epistemological picture as a matter of that picture (associated with Locke and Descartes and others) being out-of-date in a way familiar to those who have been paying attention to our discussions of the post-Kantian continental philosophers (starting with Hegel). The traditional picture asks "what is our perception of?" Do we perceive the stimuli or the objects of perception? If you admit this question, then Kuhn becomes a Kantian: surely we don't perceive the stimuli themselves; they merely cause our perception behind the scenes much as our own eyes and light rays do. So we must perceive these "objects," which must be an image or model or map of the stimuli: a representation of the real. But post-Kantians, as I have habitually interpreted them, deny this distinction between the world and the representation. As an alternate formulation, let's say that even though the stimuli get filtered and chopped up and otherwise processed through perception, our perceptions are still of those original stimuli. We are still coming into contact with those, and this is merely the way (one among possible ways) that we do it.
As with my terrain example, we are perceiving an aspect of the terrain, and if our paradigm shifts, we would perceive another aspect. Since these aspects can really be very different, claims made in one paradigm can apparently contradict those made in another, when attempting to talk about the same set of objects, because, again, what counts as an "object," what counts as part of that set, what defines all the rest of the words we use to make claims out the set, is all part of a paradigm. According to Kuhn, those in different paradigms are really talking past each other by not using their terms in the same way, and likewise the objects of their perception are incommensurable; they live in differently furnished worlds.