I regret not getting back to Thompson’s Life and Action sooner because I think I am now beginning to get the hang of the style, at least, of it.
The second section seems at first a very odd change of tack (following the first rather loose anti-reductionist discussion of life). It is a defence of what he calls a naïve theory of action explanation by contrast with a sophisticated one. A naïve rationalisation is of the form: I am doing A because I am doing B. A sophisticated one is I am doing A becauseI want to do B. The latter has a ‘movement from inner to outer, from mind to world, from spirit to nature, from “desire” to “action”’. It is also supported by a view of actin as ‘discrete or atomic or pointlike or eye-blink-like units’ . Early on Thompson advertises the fact that he will reverse the ordinary priority of naïve and sophisticated forms and hints at the analogy with barter. It is a good idea, however, to skip ahead and read the end of the essay first: the analogy on p146 and then the meat of the analysis on pp130-134. Then read pp122-128 to get more of a clue to the end that you’ve just read. Then start again at the start. (Well it’s just a thought but this does help when reading Wittgenstein and I think it might help here.)
I should stress that there’s quite a bit going on in this essay (it seems much richer than ‘Life’ (not than life!) but this is the central thread I picked out. Working backwards we have the following closing analogy:
In a so-called credit economy, after all - one in which purchase and payment are distinguished conceptually and practically - the simple act of buying something, in which purchase and payment are not distinguished, is still possible and intelligible. In a simple money economy, in which the acts of selling and buying have been distinguished conceptually and practically, a pair of us might yet engage in a simple act of barter, or of immediate exchange - an act in which the roles of buyer and seller are not distinguished. Credit and money presuppose barter in the sense that they presuppose a structuring of life which makes barter possible and intelligible, while the reverse is not the case. We can after all speak of a system of exchange by simple barter, in which such acts provide the only possible illustration of the concept of exchange. A treatment of the concept of exchange which disallows this, or which insists that every act of barter be construed as, say, the simultaneous purchase, payment, extension of credit and cancellation of debts on the part of each of two agents, is clearly absurd. My hope is to have shown that the theory of action falls into just this sort of absurdity in neglecting what I have called naive rationalization, and the sort of connection of ground and grounded that is expressed in it. 
This gives a helpful outline of what the essay is attempting to do. It displays a kind of underlying conceptual structure with the aim of ‘naturalising’ (in the good non-reductive sense of showing how something can be part of a non-spooky, albeit expanded, nature) action explanation. It is, perhaps, a Wittgensteinian object of comparison. But what’s the key de-spooking move?
The key idea seems to be this:
[T]he type of explanation of action at stake in action theory, whether naive or sophisticated, is uniformly a matter of locating the action explained in what might be called a developing process 
Hence, in naïve theory, the idea of explaining one action in terms of another and the point (raised above) that actions need not be atomic finger movements. But the verbs that make up sophisticated explanation are also derived from this idea. They are rendered less than spooky through this idea. Just as propositional attitudes have intentional inexistence, they can be directed onto non-existent objects, so in this case:
Twisting Brentano’s vocabulary, we can say that “try,” “intend” and “want” express modes of “imperfective inexistence” (of an event- or process-form) 
This idea is introduced earlier in a passage worth quoting at length:
The so-called aspectual distinction among modes of predication is easiest to apprehend if we attend to the possible formulations of what are intuitively past states of affairs involving our event- or process-forms. The English past progressive, for example, imports imperfective aspect into a proposition; the English perfect and the simple past alike import perfective aspect, in application to such verb phrases as these. Thus we can say either that I was walking to school, or else that I walked, or have walked, to school. Though contemporary action theory is bent on assimilating these propositions and the states of affairs expressed by them, the thoughts they express are of course quite unlike. That I walked to school presumably entails that I was, at some point, walking to school. But that I was, at some time, walking to school does not entail that I ever walked to school; I might have been gunned down or kidnapped by aliens, or, again, it might be that I am still walking there. The former possibility, that the truth of what is expressed by the progressive and imperfective “I was walking to school” might never be followed by the truth of what is expressed by the corresponding perfective “I walked to school” belongs, I think, to the essence of what we might call ordinary, natural or pre-scientific event consciousness, and will be of paramount importance in what follows. Though the expressions are somewhat dangerous, it helps intuition, a bit, if we say that what is registered as complete or whole or as “perfected” in “I walked to school” or “I have walked to school,” is represented as incomplete or partial or as “imperfect” in “I was walking to school.” [123-4]
So with the contrast between a perfective and imperfective aspect in play, we have also the materials for a naïve explanation. It links present and future in a way that sheds light on the present:
The use of the progressive in the articulation of ordinary event consciousness seems somehow to span the present, reaching into the future (as falling over typically includes, say, striking the ground); but the “reach beyond” the present that characterizes such thought does not expose it to simple disproof on the strength of what happens next. 
But, further, Thompson suggests that with naïve explanation as the paradigmatic deployment of the progressive, and then of ‘imperfective inexistence’, it is merely a natural development to deploy in subtle extensions the more sophisticated forms of explanation that rely on apparently more psychological verbs like try, want, intend and thus give the impression that definitive action explanation moves from within to without.
If the distinction between imperfective and perfective modes of “inexistence” of an event- or process-form can be said to be “founded deep in the nature of things,” in Frege’s sense, then “try,” “intend” and want” merely express some of the ways in which a bearer of will or rational agency can be fitted into a particular dimension of this metaphysical structure; “is ...ing,” which figures within and without the rationalization connection, expresses another. Though it acts as a paradigm, progressive judgment, as we have it, is on the present view only one form of one pole of the corresponding conceptual opposition, mastery of which is presupposed in ordinary event consciousness and the intellectual apprehension of event- or process-forms. [131-2]
And thus we end up with a picture of sophisticated action explanation as, not so much charting inner psychological depths, but articulating subtle extra joints in the idea that one action or event can be explained through its broader context.
One other bit of the essay of interest to me: the link between sophisticated action explanation and the argument from illusion. No one will need to hear anything more than this to see how this might connect to an immanent critique of that argument:
[O]ften an agent is herself tempted to give a naive account of her action, but in a legitimate third person account of the facts of her case, a corresponding sophisticated rationalization will nevertheless have to take its place. The agent may be wrong, the world may secretly be uncooperative, it may be that the agent isn't actually doing B - replenishing the house water supply, as it might be - but only thinks that she is. The general rules governing all uses of the word “because”, we said, require the truth of the propositions linked by it; P because Q, that is, entails both P and Q. And we may grant, for the purposes of this argument, that the special rules governing the employment of this connective in rationalization in particular include something in the way of a “cognition requirement”: a requirement, namely, that the agent has me grasp of the rationalizing connection, even if only inarticulately, and thus also of its terms. If, then, the explanans, or the reason, must be something the agent grasps, and if there is nothing to distinguish the cognition of the average successful agent from that of a possible parallel unsuccessful agent, then it seems that the truth that gives the ground of the action must express something that is present and grasped in either case, the successful one and the unsuccessful one. And what can this be but the agents' wanting to do the thing? So the wanting must be the true account, the real reason, and so forth. [116-7]
Now I am at a bit of a disadvantage when it comes to commenting on the essay as a whole. I’m not sure what really hangs on the debate. I can see how some conceptions of action explanation that turn of specific notions of an inner mechanism are flawed. But, given a sufficiently innocent conception of propositional attitudes, I am not sure what would be wrong with the sophisticated model, whether or not it is basic. But perhaps the way in which the third section on 'practical generality' connects back to this will make things clearer.