A number of people have already written about this book, and I doubt I'm saying anything new but here goes:
Transcritique is a reading of Kant and Marx, elucidating a common critical structure between the two which he calls "transcritique". Transcritique is the idea that every system of thought, or at least those systems which entail a judgement about the human subject and its relation to an object, contain an immanent placeholder or "transcendental apperception X" which stands in for the limit of that system or the point where it must tarry with the other. Most importantly, he does not propose that this other is absolutely Other or transcendent, but immanently other and transcendental.
The most basic way to demonstrate this philosophical stance is the emergence of Kant's thought in the antinomy of introspection as reflection. The method of interrogating one's one thoughts has long been used by philosophers, and criticized by philsophers at least since Hume. Its most celebrated practicioner is Descartes who arrived at a substantial ego through radical doubt of all empirical, external phenomena. Kant does not outright reject the idea of a unified cogito or singular point of cognition à la Hume, but instead brackets the concept and examines the "conditions of possiblity" which would allow a thinking subject to encounter itself. Karatani shows (via Kant's affinity with Husserl's eidetic reduction) how Kant shows that a person is always in three parts: the always situated empirical ego(s), the ego who begins the process of doubt which problematizes those given beings, and the transcendental ego of reflection who is implied (though never encountered) as a result.
Karatani then shows how Kant takes this same structure of antinomy to judgements which are made in the domains of science, art, and ethics. Of course each of these domains only comes into being through a bracketing of the conditions of knowledge and judgement which are necessary for the others; for example, the object of science is that which remains after feelings of aesthetic pleasure/displeasure and moral good and bad are bracketed.
Throughout all domains there is the temporal/logical knot of what Kant terms the synthetic judgement which requires them to make speculations about the infinite while acknowledging that their concept of the infinite is always a stand-in. There is always a gesture of transgression and return from the limit of every singular knowledge. Thus a work of art must assert itself as establishing a rule of beauty which is absolutely universal yet at the same time a work of genius is that which breaks with all conventions to establish a wholly new norm, every system of mathematics must axiomatically propose an infinity of situation to which it can apply without being able to finally account for that infinity, and for there to be ethical responsibility one must take responsibility for the result of one's actions, which is to say, they must act as if there was an entirely spontaneous willing where the in fact is none.
The temporality of this logic is described by the future anterior (as Zizek has rightly pointed out). That is, one must act as if one will have been free after one's action has taken place (the activity of which is not thinkable beforehand). It is here that Karatani is able to make a link with the logic of capital as described by Marx.
In commodity exchange, the value of inert goods must be decided in advance as if it had already been sold and entered into the circulation of capital. Marx showed that the two prevailing ways of thinking about economy in his day (and which persist to the present day) tried to ground value without recourse to the mediation of money: on one hand the labor theory of value which held the labor time which went into the production of a commodity as its real value and profit as theft, and on the other the theory which takes need as extra-economic and accounts for value differentially through supply and demand. Marx thus showed that the value form of money was necessary since capital relied on an excess at its core, a going-beyond.
Karatani takes an uncompromisingly materialist stance on capitalism and this ends up being both his strength and shortcoming. He deals a lethal blow to any psychologizing approach to capitalism which posits consumerism or ideology as superstructure by showing how the minimum "architechtonic" of the capitalist intersubjective relation of seller to buyer necessitates a speculative fantasy; speculation is immanent to the material of capital.
Further, he is able to proceed from the speculative fantasy of capital to the necessary ideology of the state and the nation (in much the same way that Kant proceeds through thing-in-itself, transcendental illusion, and transcendental ego as isolable only by a temporary act of bracketing). He shows how multiple other systems (the state, the nation) are autonomous but not merely superstructural, they are again indissoluble illusions which guarantee not only the stability, but the very existence of Capital as such.
It is worth going into Karatani's description of this knot. To start with, one must see that the primacy of surplus value means that capitalism has always been global capitalism from the start. This is because surplus value is created from the discrepancy between systems of value. In the case of merchant capital it is the difference of price for the same commodity in different places and in industrial capital it is the difference produced by technological innovation. In either case it is not a simple matter of price difference and profit, it involves an exchange with a future or other system of value which can only be presumed to exist and which is accesible only to the value form (money, gold, or whatever holds its place). However, for capital as value form to exist, this difference cannot merely be oppurtunistic theft through mechanical or geographical advantage, but a result of a total system of differences. This is a point which Karatani repeats many times in the Marx section: for capitalism to function and thus for Capital to exist as such, the workers must buy back the commodities they create in totality , or else there would be economic crisis. Thus, a government which functions as a regulative instrument which forcibly takes money and redistributes it to the whole is necessary for the internal inscription of difference into the capitalist system.
Nation proceeds from State in an entirely analogous way. If the state in capitalism functions as a representative of totality, as that which is not directly represented by short-sighted capital but which is nonetheless its fundamental condition of possibility, then nationalism is the form of power necessitated by the inability of state to represent the masses. That is to say, the form of power proper to capitalism is representative democracy as the image of the bourgeoise class in their capacity of possesors of value-form, but that this mechanism of representation cannot but fail to account for the proletariat as those who exist without having any share of value-form, who form a class only in that they do not share in the generality of money, a class-which-is-not-one. They are, however, people and thus must be represented in universal sufferage though they cannot represent themselves. What must act as their representative is that which is transcendent to the system, the bonds of nation and fraternity which have as their representative a sovereign who is himslef above the law while remaining a part of the beauracratic capitalist-state (it is not, as Karatani stresses, a regression to a feudalism but the persistence or return of the repressed feudalism within the thouroughly modern network of capital/nation/state).
The problem with Karatani's book is his refusal to take his project far enough, his timidity at universalizing his method. He does not bring Kant transcendental subject back to bear on the contemporary capitalist subject. Instead of being a function of subjectivity as such, surplus value is seen as wholly persecutory and thus extra-human in origin. This is why he is able, in the end, to revert back to a relationship of "association" as exchange without surplus as the eventual overcoming of capitalism. If he stays true to his original approach he must arrive at a theory of subjectivity which places the object-embodiment of lack/surplus at the very core of the subject. By doing so, however the possibility of real revolution would again have to replace the nonviolent, passive resistance of boycotts and assoctionism.
This place would be the pure action of doubling by which a Subject declares himself as identical subject and thereby allows object-of-difference (surplus/lack) to fall away. For Karatani, plus-de-jouir is Capital and Capital is plus-de-jouir. Thus, the only way to escape capitalism is the elimination of excess as surplus value which he believes is achievable through the Local Exchange Trading System. There is no room in his theory for a creative use of the reemergence of surplus value which takes place in revolution.
The reemergence of object is possible through the declaration of outright revolution as spontaneous break with all current systems of representaion and the subsequent deliberation which utilises the difference announced to create previously unthinkable forms of exchange and government. Instead he relies on a form of exchange which is only ostensibly new (association, even exchange) but which is actually the fantasy of a capital without production or exploitation which is so rigorously decried by Karatani throughout the rest of the book.