The answer to the question in the title is this: the cat has to be able to let in a tiny amount of light into their eyes during daylight hours as their eyes are more sensitive to light than those of a human.
When a cat’s pupils are fully contracted, leaving a narrow slit, only a tiny amount of light enters the eye. The cat is able to further restrict the amount of light entering the eye by lowering their eyelid over the slit as if a curtain is being drawn downwards.
The reason why a cat has very sensitive eyes is because they hunt at dawn and dusk and mainly during the night. Domestic cats also hunt in the daytime. However, we all know that cats are able to see much better in the dark than humans.
Humans do much of their activity in daytime therefore their eyes are suited to the amount of ambient light available during daytime. The eye is less sensitive. When the pupil is fully stopped down, to use a photographic term, or fully contracted, it still lets in more light than the eye have a cat. Read an article on comparing human and cat eyes.
The cat’s eye is sensitive to light because there is a reflective layer (tapetum lucidum) behind the retina which reflects light back into the retina resulting in a greater ability to pick up light and therefore to create images.
Some exotics cats too only have round pupils :O)