The microwave wave guard cover is part of the overall cooking chamber. The cooking chamber itself is a Faraday cage, which is an enclosure designed to neutralize the microwaves emitted by the magnetron within the microwave oven. The wave guard cover has small holes that allow you to view the food but still shield you from exposure to microwaves. This is because the wavelength of visible light waves is smaller than that of microwaves; the holes are large enough for visible light waves to pass through but too large for almost all microwaves to pass through.
The U.S. Food and Drug Administration limits the amount of leakage of microwaves from a microwave oven within its lifetime to 5 mW per square centimeter, at approximately 2 inches from the oven surface. The microwave wave guard cover, as well as the whole microwave cooking chamber, ensures that requirement is met.
High levels of microwaves are known to cause health issues, such as severe burns, temporary sterility or cataracts. These problems are associated with exposure to far higher amounts of microwaves than the microwave wave guard cover and Faraday cage are allowed to leak. In addition, the strength of the microwaves radically decreases with the distance from the microwave; for example, at 20 inches from the wave guard cover, the microwaves are 100 times less powerful than at 2 inches. There is little information on the effects of long-term exposure to low levels of microwaves.
You are not advised to attempt to repair your microwave wave guard cover on your own. The chamber that houses the guard cover is designed to be free of any particulates such as dust or dirt, which could interfere with the wave guard cover's ability to neutralize the microwaves; home repair is likely to introduce such particulates. Also, you cannot replace any part of the guard cover's metal perforated sheet with any household metal sheeting because, again, the guard cover is specially treated to neutralize microwaves.
Your mother may have told you to not to stare closely into the microwave because you would absorb harmful radiation. This is mostly false. It is true that the closer you are to the microwave wave guard cover, the stronger the microwaves will be, if any have leaked at all. However, the FDA-required limit on leakage is far lower than what is harmful. Also, the radiation from microwaves is not cancer causing, which may also have been a cause of your mother's worry.