Like all of Deleuze and Guattari's concepts, this is one that can be interpreted in a variety of ways, and is manifested on a variety of levels. It's most central to their collaborative work in Capitalism and Schizophrenia, though Deleuze used it decades prior, and the phrase comes from a radio play by Antonin Artaud. In brief, the body without organs (or BwO) is best understood as their way of conceptually emphasizing dimensions of embodiment beyond organization, or in other words of looking at how different kinds of bodies become organized, rather than thinking of them as static, already-organized wholes.
From his very first book, Deleuze was fascinated by the field of embryology, which is nothing other than the science of how organisms acquire their organization. And eventually, with Guattari, he will describe the BwO as "an egg," a substance "crisscrossed with axes and thresholds, with latitudes and longitudes and geodesic lines, traversed by gradients marking the transitions and the becomings" (Anti-Oedipus, p.21). This has to be understood in relation to how science and philosophy have traditionally conceived of the body. In anatomy, the privilege has always been on the normal, healthy, functional organ, to be understood individually as a definite entity but also conceived as part of an integrated organic whole. In a more philosophical key, this leads not only to the idea of the body as a well-oiled mechanism under the control of a transcendent mind, but also the idea of the 'social organism,' of society as an integrated whole made up of differentiated functional components, again under the control of some sovereign authority.
By shifting the emphasis from the organism to the Body without Organs, Deleuze hopes to accomplish two things. First, to demonstrate the becoming of the organism, the ways in which differentiated organs are formed out of an undifferentiated mass, along 'axes and thresholds' of development. Thus he wants to suggest - by drawing parallels with embryology and sociology - that past philosophers have misunderstood this dynamic becoming through categories of static, organic being. When being is conceived in terms of static elements or 'organs,' however, their coming-into-being becomes a mystery, and seems to demand some transcendental 'designer' - be it God in the case of organic forms themselves, consciousness in the case of the forms of thought, or some political sovereign in the social realm. And so the second aspect of his move is a critique of what he calls the 'hylomorphic schema' common throughout the history of philosophy, of a 'passive, inert matter' waiting to receive a form from somewhere or someone else. For Deleuze, when conceived as an 'organism,' bodies call out for an active designer who might impose a form on their brute matter from outside; the BwO is instead a way of privileging immanence, the self-organizing, form-generating capacities within matter itself, requiring no transcendent authority.
In this regard, Deleuze and Guattari refer to things like the capacities of metals to enter into certain alloys, of organic molecules to fold in certain ways, of social groups to organise themselves (and really one has to look more closely at D&G here for the details, along with Manuel de Landa's scientific 'translations' of their conceptual vocabulary). And while the kinds of self-organizing processes they're interested in are most prominent in the embryological or 'larval' phases of development, they view even organized bodies as drawing from a BwO, in the crisscrossing networks of flows that keep any dynamic system active: blood and neural firings in a biological body, for instance, or money and ideas in a social body. (Or solar energy on the body of the Earth, for that matter - this comes into play at all scales.)
The final point, then, is that for them the question of 'interpreting' the Body without Organs isn't quite right; for them the question is always an active one, "how to make yourself a body without organs." The answer is whatever brings you closer to those flows that sustain you, in effect the pursuit of intense embodied experiences (both in the common-sense meaning of the term and Deleuze's own special sense of intensive). So in Capitalism and Schizophrenia they laud schizophrenics, drug users, and masochists, with their bizarre depersonalizing rituals and monstrous fantasies; but on my reading you can make your BwO however you like, whether you like being whipped or doing whippets, or instead you prefer day-trading, political activism, or marathon running.