Translation:The bugs are crawling off of the newspapers from which we are reading.
You could give various arguments for that, but formost I'd like to remind you that vmiről also has the meaning of 'about'. We could read about newspapers from which bugs are crawling.
The other argument I like to use it that written pieces have content, and contents are usually found inside of things. You read and write in books and newspapers in English, too.
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That is what the "Az..., amelyik..." construction expresses. There are a lot of newpapers around, but bugs crawl only from those that we are reading. But I think in English it suffices to say "the newspapers that/which we read" to express the same thing.
You can, however, modulate the meaning a bit by saying "A bogarak lemásznak az újságokról, amelyekből olvasunk." That puts the newspapers and attachment a bit in the background. The base word of amelyekből here is amely instead of amelyik, and it can be used without the az construction, but instead refers the last noun before the comma. vvsey has written a wonderful entry about that somewhere, but I fail to find that right now. You might stumble over it somewhere.
I wouldn't go as far as to say it's nonsense - I'm pretty sure I know what they're saying - but the wording isn't using naturally spoken English. "The bugs are crawling off (of) the newspapers (that) we're reading." "Of" and "that" are in parentheses because I don't think they're completely necessary, but that might depend on which dialect of English you speak.