So light from the Sun is made up of all different colours. When this multicoloured light hits the atmosphere, and all those pesky particles get in the way, some of it gets scattered about. Now the shorter the wavelength of light, the more likely it is to get scattered; and blue has a shorter wavelength than other colours. So it gets scattered more.
This means that most of the colours of light will pass by and carry on to wherever they were going, but a higher percentage of blue light will get scattered and end up heading towards your eyes; which makes the sky look blue.
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