He doesn't mean to say the conversation is meaningless, just that the signs themselves have no intrinsic meaning, since they're arbitrary. Which, duh. It's a banal claim articulated poorly.
There's a little bit of ore here to mine (there are philosophical accounts of natural meaning, for example; see Paul Grice), but he mostly seems to just believe in a transcendental signified, and missed the boat on how incredibly rich the consequences of e.g. J. L. Austin and late Wittgenstein are for how we think about language.
Grice and Wittgenstein are good starts, but Cassirer, Basil Bernstein, Halliday, Reddy, Lakoff, Winograd, Stitch, Ryle, Givon, Deacon, Fontaine all weigh in on the notion that language doesn't cohere meaning. Each offers a different variation on whether any symbol, metaphor, grammar actually holds or refers to semantic. When animal signals are considered, we get to see specific signals in action (Rendall, Cheyney, Hauser) and it's here the conduit metaphor's glaring paradox becomes evident, even for speech. If an utterance can never generate the same or extremely similar metal state that generates an action repeatedly, then the utterance is arbitrary, it's for all better purposes meaningless.