Maybe some surface oxidation of the conductor occurs (which will make the sound slightly less bright. Perhaps some charging of the dielectric insulation occurs. I would not discount the possibility that the (measurable) characteristics of the cable change, but I'd like a proponent of the burn-in theory to show me their measured results.
However I think it is more likely a psychological effect : the brain rapidly adapts to new tonal balances, and comes to like them. I have a hifi in England at my parents' house and one in the US both of similar quality. I have noticed that on my vacations in England in the first few days my UK system sounds worse than my US system. Then when I return home the first few days after I get back my US system sounds worse. Over the long term I am very happy with the sound of both systems ... they're just tonally different.
However I think it is more likely a psychological effect : the brain rapidly adapts to new tonal balances, and comes to like them. I have a hifi in England at my parents' house and one in the US both of similar quality. I have noticed that on my vacations in England in the first few days my UK system sounds worse than my US system. Then when I return home the first few days after I get back my US system sounds worse. Over the long term I am very happy with the sound of both systems ... they're just tonally different.