The Sun (which is a star) is so bright that it overpowers the light from all the other stars put together. Although the Sun is not the brightest star in the universe it is the closest and that is why it appears so bright. If other stars were much closer to the Earth you might be able to see them during the daytime.
There is one occasion where one can see stars during the day, namely during a solar eclipse.
This fact was exploited by Arthur Eddington in 1919 to check that light rays from stars nearby the sun were bent by the gravitational effect of the sun.
As Stephen points out, one can only do this if one blocks out the sun. This constituted a beautiful experimental check that Einstein’s theory was correct and made Einstein a household name.
If you had some way of filtering out the Suns light then you could see the stars during the day. The Suns light just overwhelms everything else in the sky (except sometimes the moon can still be seen). During solar eclipses some of the stars become visible for a short time.
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