Unfortunately for this claim, if you go to the book cited, what you actually find is a calculation of the number of possible roots in Arabic, without regard to whether or not the root actually has a meaning. Such a count includes huge numbers of unused roots such as بزح bzḥ or قذب qḏb, while at the same time lumping together all words derived from the same root; كتاب book, كاتب writer, and مكتب office are three words, but only one root. The result of such a calculation might tell us something about the potential for expanding Arabic, but absolutely nothing about the state of the Arabic language. And since in practice both Arabic and the languages it is being compared to on that poster allow arbitrary long words without real roots, if only in loanwords, it doesn't even tell us much about its potential.
Both the number of Classical Arabic roots with actual meanings and the number of words can be estimated from the classic dictionaries: according to Sakhr's statistics, there seem to be around 10,000 roots, and up to 200,000 distinct words. Roots don't play such a major role in the lexicography of most non-Semitic languages, so it's difficult to compare the number of roots cross-linguistically. But in terms of words, that would be slightly fewer than English (250,000 in the OED, although the poster cites 600,000) and slightly higher than French (over 100,000 excluding proper nouns, according to the Académie Française).
However, such comparisons can hardly fail to be misleading. For one thing, English is much more hospitable towards dialectal and colloquial usages than Arabic is – the OED is full of words marked as Scottish or Northern or slang or whatnot, the equivalents of which would never be accepted by an Arabic dictionary. For another thing, the whole enterprise of counting words across languages runs into apparently insuperable problems, especially when it comes to compounds, which Arabic dictionaries do not normally treat as words. If you include compounds, then compound-friendly languages like German or Turkish or Inuktitut are automatically going to beat all the rest – and all the available statistics that I've seen for, say, English happen to include compounds.
So the best answer is that we don't really know, and that word count, even if we could measure it better, is not a very good measure of a language's expressive power anyway. Some missing words make a genuine difference, as I've discussed here before. But is English really missing out by not having distinct words for male camels (جمل) vs. female camels (ناقة)? Is Arabic really missing out by not having a special word for cornpone, or for scones?