Translation:The politician who is talking on the radio is famous.
As best I can tell, English makes it clear that you are qualifying the politician because the relative clause "who is takling on the radio" is right next to the word "politician".
But in Hungarian, where the relative clause is (at least in this course) often not right next to the noun, az is used as a signal, much as ott or the like are used as signals in other sentence to indicate places or the like, where English would not use them (we'd say "I play where you do" and not "I play there where you do").
This signal az does not need to be translated.
But it could also mean "That politician...", as there would not be a different way to say that in Hungarian.
You are probably right about "this course". But in real life, at least for me, I would make this distinction:
"Az a politikus híres, aki a rádióban beszél. " - "That politician..."
"A politikus, aki a rádióban beszél, híres." - "The politician...".
See, there is no problem putting the referring clause right after the subject.
And you can even say:
"A rádióban beszélő politikus híres." - "The politician talking on the radio is famous."
So, I would say, this course either reflects somebody's curious personal style of speaking Hungarian (very well possible) or we are dealing with a serious mistranslation of these kind of sentences throughout the course.
"Az a híres politikus, aki a rádióban beszél."
It might be easier to decide between "that is the politician" and "that politician is" if you remove the dependent clause: "Az a politikus híres." The adjective comes after the noun it refers to (it's in a predicative place), so it has to be "That politician (is) famous". Hungarian and English work the same in that regard. :)