Why do anticyclones bring clear skies in summertime?
Why anticyclones are also called areas of high pressure

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. 

Impact of high pressure areas on weather

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.

Background info on clouds and heat

Heat generated by the sun travels two ways:


Towards Earth: From the sun, through space, hitting the earth
Away From Earth: From the earth and back out into space


Clouds reflect heat. 


Either away from earth, as incoming sunlight hits the tops of
clouds, or            
towards earth, as heat radiating away from earth's surface towards space hits the bottom of a cloud and bounces back


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.