It's obvious that in summer anticyclones bring hot weather. But why clear skies?
Heat evaporates the moisture and should lead to the cloud formation, shouldn't it?
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Let's call anticyclones areas of high pressure.
Areas of high pressure (anticyclones), pull air towards the earth's surface. The increase in the number of air particles near the earth's surface causes a higher pressure for things on the earth, hence the name.
High-pressure areas don't always bring hot weather.
High pressure areas do prevent the chances of clouds forming. Why? Because more air is pushing down around the earth's surface, rather than rising up in the air where it can cool and form clouds. This is why areas of high pressure (anti-cyclones) bring clear skies.
Because high-pressure areas prevent clouds forming, this can cause hot fine weather in summer when the temperatures are warm and sunny days are long, or cold weather in winter, when sunlight hours are short and the lack of clouds means any warm can escape from earth's surface and back out into space.
Heat generated by the sun travels two ways:
Clouds reflect heat.
Areas of high pressure stop clouds from forming, which can mean warm summer days or cold winter days for the reason above! (In summer, a lot of sunlight can get through to the earth's surface, in winter when there's little sun, any of the earth's heat escapes into space with no cloud to stop it!)
I presume you are a beginner, so I've given a basic and easy to understand answer. Once you wrap your head around this, I can update with more details.