The stars don't all appear at the same time at night, either. You can see the brightest stars first, because then the sky is already darker than the brighest stars, but still brighter than the dimmer stars.
If there weren't any air on Earth, then the sky wouldn't be blue, and then you'd be able to see stars in the sky, if you didn't look at the Sun, at least, because you cannot see a small light well if it is very close to a very bright light.
There is no air on the Moon or in space, so from there you can see the stars during the day.
On Earth, you can still see stars during the day if you use a very big telescope (so there's only a very tiny patch of sky in view next to the star) or if you experience a total eclipse of the Sun. During a total eclipse of the Sun, the Moon blocks nearly all of the sunlight. Then it gets dark as if it is night, and then you can see the brightest stars.
Scattering of light happens also at night. Of the light of streetlights, greenhouses, cars, and houses that goes up into the sky, the air scatters a small part back down again. That's why from a city or an area of greenhouses the dimmest stars or the Milky Way aren't visible even on a clear night. That's also why astronomers take their telescopes to faraway places where there aren't any towns or greenhouses nearby: then the sky is still really dark at night and then you can see many more stars.