A radical is a thing that, or is a person who, goes to the root of a thing. Since radix means "root," that is what a radical is. Stem, branch and leaf do not much interest the radical, but only the root from which the others have sprung.
If the radical is a person, then the word often (but not always) carries a pejorative connotation that the radical is one who unreasonably entertains a simpleminded caricature of the root; that the radical's knowledge of the root is shallower than he or she believes or is willing to admit.
You probably knew that already, but the point is that the sentence you quote is not very good English.
One specific use of the word radical that has grown prominent in the English-speaking world during recent decades indicates a Muslim who would shear away centuries of Muslim history and culture to revive a primitive, early, putatively pure understanding of Islam. The inference is often implicitly associated to unpleasant behaviors like beating wives and blowing innocent strangers up. That's a lot of recent connotation to load into an innocent old word like radical.
The fact that one uses (or that I speak of) the word radical in this way should not be taken to imply that the user (or I) actually knows very much about Muslims and Islam. I merely explain a frequent way in which the English word radical has recently been used.
A good writer would not use the word in that way unless having first established a clear context thereto.
As far as the phrase "radicalized against the country" goes, well, one vaguely gathers from the context that (i) the country and/or the radicals are Muslim, (ii) the radicals have been behaving unpleasantly, and (iii) the writer suggests that it is reasonable to suppose that a malign influence of radicalism is what has caused the radicals to behave unpleasantly. But, really, the writer seems to be confusedly mixing images associated with the aforementioned recent use of the word radical with the factual elements of whatever story he or she is reporting. It is hard to tell what this writer means. He or she may merely mean that the state has made bitter foes of millions of Yemenis. Because Yemen is a Muslim land, the word radicalize is limply tossed in there for flavor—and, no, you are right: usage-wise, that makes no sense.
The writing is bad, so if my interpretation is wrong or yours is wronger, then we can blame the writer for that.
Good question. Indeed, the question is much better than the quote.