When the air hits a slow moving part, it piles up on itself and sinks to the ground causing an area of high pressure. As the air sinks it squashes together and is compressed, causing it to warm up. This is due to one of the simplest laws of gases, which is if you heat a gas it expands, but if you compress a gas, it will release heat. So a high-pressure system will bring warm air. But this is not all.
Warm air can hold more water vapour without it turning into cloud.
Thus not only will high-pressure bring warm air, it will also remain fairly cloud free meaning the weather is sunny and bright. Low-pressure systems occur when the air in the upper atmosphere is moving fast, so fast that it spreads itself out and the pressure drops. This causes air to rise from below into the upper atmosphere, resulting in an area of low pressure. As the air rises from the ground, it expands into the low-pressure area above. Just as compressing gases makes them to warm, allowing gases to expand and spread out causes them to cool.