That doesn't make any sense. In fact it makes non-sense. How can 'tradition' 'explicitly' prohibit anything? It can't: that's why it's called tradition to distinguish it from law. Law can, obviously, explicitly prohibit things, but tradition can't, it can only implicitly prohibit them. Tradition isn't written down or codified; it's fuzzy; it's implicit; it has blurry edges. It's the very opposite of explicit.
But that's not exactly true. Tradition is just the handing down of doctrine or life-shaping principles; there's no reason to think we can't have written traditions. The most obvious example would be the Jewish Talmud, but in fact Islam has its own good example of written tradition, namely, the ahadith. A hadith is a written tradition about the Prophet (something heard by the writer from someone who heard it from someone who was there, for instance -- this chain of transmission or tradition is called the isnad). While such traditions do not have the authority of the Qur'an, given that the Qur'an enjoins respect for the prophets, they have played a major role in shaping Islamic perspectives, and much of the diversity in Islam comes from the fact that not everyone accepts the same ahadith as authentic. (In fact, which ahadith a Muslim scholar accepts as authentic will tend to have an immense influence on which school of Islamic jurisprudence he prefers, and vice versa. Ahadith are a major part of the data from which Islamic law is derived.) Through the ahadith one gets a picture of the sunnah or way of life of the Prophet. And clearly a tradition in this sense can explicitly prohibit or require; e.g., a tradition that the Prophet forbade pictorial representations of a certain kind.
And, as it happens, the BBC is repeating itself here. In a Q&A in February, we find this sentence:
Islamic tradition or Hadith, the stories of the words and actions of Muhammad and his Companions, explicitly prohibits images of Allah, Muhammad and all the major prophets of the Christian and Jewish traditions.
In fact, although I haven't looked at all the relevant ahadith, it seems to me that the ones I've seen put forward allow for a little more interpretive room than this suggests, and, of course, they are in any case not found in every collection of ahadith, nor are they accepted by everyone; but it's clear that the intention is to take 'Islamic tradition' as a sort of paraphrase-translation of ahadith. Benson is right that it's a bit misleading -- but what makes it misleading is not that tradition can't explicitly prohibit anything (which is obviously false since you can have even an oral tradition that carries forward an explicit prohibition, as with a taboo or an explicitly taught social norm), but instead that it makes hadith sound more monolithic, and less disjointed and patchwork, than it is.