Kant sometimes contrasted appearances with "things in themselves," which in turn are sometimes equated with objects of the intellect, noumena. Schopenhauer understood the notion of things in themselves to be metaphysical, that is, to indicate a kind of reality, one more fundamental than that of appearances. (This interpretation is hotly debated in the Kant literature, where many commentators claim that the appearance/thing-in-itself distinction concerns only a way of considering objects.)
According to Kant, things in themselves are unknowable, rather than objects of intellectual knowledge. Moreover, appearances have the structure they do through the filter of human cognitive faculties: the sensible faculty orders appearances in space and time, while the understanding conceptualizes it, most importantly as standing in a causal order. Shopenhauer maintained that the subjectivity thus introduced into the description of appearances placed Kant in the camp of idealism, along with Berkeley.
But this introduced a puzzle, since in the Critique of Pure Reason, Kant vehemently denied that his philosophy was Berkeleyan at all. There were also many passages which seemed in conflict with the fundamental idealistic tone of the work. Schopenhauer's initial response was that Kant was simply inconsistent. But upon his discovery of the first edition of the Critique, he found a solution to the puzzle: Kant's statement of his idealism was quite explicit, the conflicting passages having been added in the second edition. (Schopenhauer attributed this retraction as having been driven by fear of being labeled a Berkeleyan.)
But "pure" Kantian idealism, as interpreted in the penultimate paragraph, still had problems of its own. Schopenhauer noted that Kant introduced the notion of things in themselves as the source of "impressions" which are "given" to the mind. It is these impressions which are ordered by space, time and causality. But as an early critic (G. E. Schulze) noted, Kant here runs into an inconsistency. The category of causality is not supposed to be valid for anything but appearances -- a point Kant made repeatedly. Such "transcendent" application of the category is precisely the source of the illusion of knowledge of noumena.
Insofar as Schopenhauer thought of his own metaphysics as a simplification and extension of Kant's, he had to find an answer to this objection. In the first place, he rejected Kant's claim that things in themselves are unknowable. The claim is of little use, given that Kant had violated his own strictures by in effect claiming knowledge of things in themselves as grounds of appearances. As we have seen, Schopenhauer uses the term 'will' to refer to things in themselves.
But Kant was right when he held that we have no intellectual knowledge of things in themselves, according to Schopenhauer. Our knowledge of will comes not through concepts but rather directly, through self-consciousness. We are conscious of ourselves as objects in space and time among other objects. But in acts of willing (which appear in space and time as motions of our bodies) we find the unvarnished reality behind the human body. As bodily motions are identical to acts of will, the body itself is an objectification of the will.
In the second edition of The World as Will and Representation, Schopenhauer illustrated the way in which we catch glimpses of the underlying will. We are conscious of ourselves as intellects, capable of abstract representation. Some of our representations are ends which we desire to achieve, and they serve as motives for our actions. Will cannot be found in consciously motivated behavior, but it can be detected when our intellectualized motives are overruled by something more powerful and basic, i.e., by the will.
This occurs, for example, when we express delight at some event which is quite contrary to our conscious desires. The pleasure felt at the unexpected death of a rich relation I believed I loved reveals an inner drive which I had kept hidden from myself. It should be noted that the doctrine of a hidden, unconscious source of motivation later became highly influential. Prior to Schopenhauer, it was generally held that the contents of our inner selves are "transparent," rather than in any way hidden.
A problem for this view, however, is that as described, my will seems to have its own ends, e.g. my enrichment. But ends are possible only insofar as there is an abstract representation thereof. The will as thing in itself knows nothing of the represented world. In Schopenhauer's own figure, it is like a blind giant carrying a sighted dwarf.
This difficulty is really of a piece with Kant's imputed problem with his description of things in themselves as sources of the given. If the intellect is confined to the sphere of appearance, there can be no conceptual description of things in themselves: they can only be "felt" immediately.
Another potential problem with the doctrine of will as thing in itself is that for each person, the immediate relation to the will stops at one's own body. A skeptic would ask why anyone should think that other perceptual objects are objectifications of will. A theoretical egoist (solipsist) goes further, claiming that nothing but himself is real.
Schopenhauer's reaction to solipsism is somewhat typical, though not very satisfying. Theoretical egoism cannot be proven (how could one show that there is not a reality behind the appearances?). And we can safely ignore egoism because the egoist cannot press a case against anyone else, locked up as he is within his own world. This calls to mind John Locke's response to those who are skeptical about the existence of an external world: "At least, he that can doubt so far, (whatever he may have with his own thoughts,) will never have any controversy with me; since he can never be sure I say anything contrary to his own opinion" (Essay Concerning Human Understanding, Book IV, Chapter XI, Section 3).