Translation:He can speak many languages.
I'm not a native Chinese speaker, but this is how I understand it:
会 means that someone knows how to do some kind of skill. 能 means that someone is physically able to do something. So, for example, 我会开车means "I can drive (a car)" (as in, I've passed my driving exams, I know how everything works, etc.). 我能开车 means "I am able to drive (a car)" (as in, I'm not drunk at the moment, so I'm physically capable of driving). I've heard from some people that actual Chinese speakers sometimes don't care about this distinction, though.
The exercise sentence could be explained as "he is physically able to speak many languages." This feels strange to me, because I'm assuming the intended meaning of the sentence was "he knows many languages", not something like "his tongue and mouth are working properly, so he's capable of speaking many languages." Again, I guess the technical differences between 会 and 能 are ignored sometimes.
Nice explanation! Its not easy to explain. The short answer is that it feels "right" to say it one way and not the other, so that the only way to learn how to "feel" is through more exposure to the language (reading, listening, et cetera), and the long(-er) answer is that in this case, 会 implies that he, or she, can speak but with an undisclosed level of fluency whereas 能 seems to imply that he or she is at least reasonably fluent in those languages. Again, this is difficult to explain. Some people might even say the opposite is true, but this is from me; feel free to contribute to this discussion.
能 is about having an ability. 会 is about being able to do something. If I have the physical ability and skill to swim 我能游泳。If I am currently forbidden from swimming 我不会。
The sentence here considers being able to speak a language an ability. If this man got laryngitis 他还能说很多种语言，可是现在他不会。
Oh, actually, that is rather the opposite, for the examples, at least. If you are forbidden to do something one usually says 我被（医生，perhaps？）禁止游泳，所以暂时不能游泳，也不会去游泳。
As to the sentence here, and the situation you raised, the last part needs to be changed, and one answer is 可他现在暂时不行. It is just the last character, really, but that is an alternative structure.
Yeah, that's an interesting question. Since 种 is the classifier for both languages and kinds/types of things, I'm assuming that Chinese doesn't make an explicit distinction between "X languages" and "X kinds of languages." What the speaker means could just be determined by context.
I went to Google translate and typed "five languages" and "five kinds of languages", and both were translated as 五种语言.
This is an example of the measure word 量词 overlapping with other meaning(s).
In Chinese, 量词 is used much more often than measure words are in English. For example, "a boy" would be translated as ”一个男孩“, "three strawberries" as ”三颗草莓“ and so on.
There are other cases where there are other overlaps too, and one example that comes to mind is 五层楼 for "five stories", most people just say 五楼，e.g. 这座图书馆有五层楼，设备齐全，环境舒适，是个读书的好地方。
This roughly translated to "This library has five stories. It has good (comprehensive) facilities, and a comfortable (working) environment. It is a good place to read/study/work on some reading."
Chinese uses a lot more commas than English; that is another, punctuation-based difference to note.