Why would a factory put on signs on the site that people cannot take pictures on site without permission from the general manager?

It is not unusual for plants and factories to not allow photos inside the facilities. It is partially for security and partly to protect trade secrets.

There could be equipment or processes that are patented or which give that plant a competitive advantage in the marketplace, or there could be pre-production / future release parts or goods which are confidential within a company and whose release to public / competitors would harm them.

ah -- it may not be hide so much as preventing others from profiting from the company without payment. by prohibiting photos, the company asserts that all photographic rights belong to it.

note that any company with any sort of sensitive work, or Defense contracts, is required to do this [and much more in the way of secrecy] by terms of their contract.

Industrial espionage perhaps.