How valid is the claim that the Renaissance came about as a result of monks translating the works of Muslim scholars into Latin?

It is wildly inaccurate. There are several ways you could interpret the "Renaissance", some meaningful, some not very meaningful, but for the purpose of this question I think the most relevant would be: the "Renaissance" was a widespread reawakenment of interest among western Europeans in Greek antiquity (especially among Italians), coinciding with the introduction of the printing press and heavy importation of Greek books from the Byzantine world.

Now, the story you report in your question is to do with events prior to the above. Greek books certainly got into western Europeans' hands before the "Renaissance", though at a much slower rate, and there were two main ways that they did so. One way was direct imports from the Byzantine world. The other main way of importing Greek books was in Arabic translations, especially in Spain.

After ~~1500~~ 1400, the second method ceased to be very relevant any more, because the pace of imports from Byzantium increased drastically, firstly thanks to individuals like Giovanni Aurispa and Cardinal Bessarion, later thanks to the fall of Constantinople. So really, the Renaissance marks the point where western Europeans stopped being so interested in Arabic translations, because it became easier to get hold of the originals that the Arabic translations were based on.

This way of thinking about the Renaissance is almost purely the history of the transmission of ancient Greek texts, so if you want to read more, that's the topic to look up. My personal favourite book on the subject is Reynolds and Wilson's Scribes and Scholars, which covers both west and east. But there may be some books out there with a narrower focus on Greek transmission.

Since the Renaissance, our knowledge of ancient Greek texts no longer depends on Arabic translations -- with a few exceptions (there are a handful of philosophical and mathematical texts that survive in Arabic, but which didn't survive in Constantinople). I believe the idea that they still matter is a myth invented by Carl Sagan's Cosmos, though it may be pre-Sagan too.