The organs are the sense organs mentioned at the top of the passage. Nietzsche mocks sensualists, who insisted since antiquity that "every thought comes from sense-perception" (Sextus Empiricus), "there is not anything in mind, which hasn't been in the sensations" is even sometimes attributed to Aristotle. Closer to Nietzsche's time the torch was carried by British empiricists like Locke, Hume and Mill, but especially Berkeley, who declared that esse est percipi, to be is to be perceived.
Still, Nietzsche's mockery of "external world is the product of our organs" exaggerates even him. What British empiricists actually claimed, contrary to rationalists like Descartes and Leibniz, was that mind has no "innate ideas" or "rational intuition", and that all its content is the product of processing sensations. Apparently, Nietzsche's did not take kindly to the idea, but his "reductio" of sensualism does not work of course, and he suggests it half in jest. Technically, he equivocates on the word "product", sensualists use it epistemologically, while Nietzsche needs ontological meaning for his "reductio".
It is interesting that on causality Nietzsche fully sides with Hume's sensualist treatment against Kant's a priori, and "sense organs are not appearances in the way idealist philosophy uses that term" is a stab at Kant, who sharply separated "things in themselves" from "appearances", and refused to grant the former status to anything sensual.