The sentence you quoted uses neither to implicitly continue the use of allowed from the first sentence to the second sentence.
Neither is used to negate two subjects, so you could say "cameras are not allowed, and neither [allowed] is making copies". An equivalent expression would be "cameras are not allowed, and making copies is also not allowed". Thus, you can replace neither with also not allowed in the original sentence:
"No cameras are allowed. Also not allowed is making copies of anything."
The second sentence is written in an inverted form, so you could rearrange it by placing the subject, making copies of anything, at the beginning to make it sound more natural or understandable.
"No cameras are allowed. Making copies of anything is also not allowed."
Here is my best guess as to why the original sentence was written in this way:
To use neither to refer to both subjects in just one sentence, it is best to use two singular subjects or two plural subjects:
"Neither cameras nor phones are allowed." → Cameras are not allowed, and phones are not allowed.
"Neither printing nor making copies is allowed." → Printing is not allowed, and making copies is not allowed.
Since the original sentence has one plural subject and one singular subject, it would sound incorrect to use neither to refer to both in a single sentence.
"Neither cameras nor making copies are allowed." → Cameras are not allowed, and making copies are🚫 not allowed.
"Neither cameras nor making copies is allowed." → Cameras is🚫 not allowed, and making copies is not allowed.
I would guess that the writer chose to separate the two subjects, cameras and making copies of anything, and use neither is to refer only to the second subject, in order to avoid this subject-verb disagreement.