Why does high-pressure spin clockwise in the Northern Hemisphere?

The reason is simple: it is due to the Coriolis Effect.

This effect is caused by the way objects move in “straight lines” in a rotating reference frame. Here is a Wikipedia gif that I love using to help explain the effect:

From a north polar perspective, which is appropriate in the northern hemisphere, all moving objects are deflected to the right as they move, which gives them a clockwise trajectory on the surface of our counterclockwise rotating planet (from the northern perspective).

Now, meteorologically, and in the simplest terms, high pressure attempts to move toward low pressure in the atmosphere. Having areas of extremely high atmospheric pressure right next to areas of very low pressure is unstable- they like to neutralize and air from the high-pressure system will flow outward from it. As it does so, it behaves like any other object moving along the surface of a rotating object. Thus, the high-pressure system will have a right-hand spin in the northern hemisphere- it rotates clockwise.

A low-pressure system, rather than having air moving out of it, will have air attempting to enter it for the same neutralizing purpose. That air will also be deflected right, but note that this deflection, because the air is moving in the opposite direction to that of the high-pressure system, will result in a counterclockwise rotation.

Also note that from a south polar perspective, which is appropriate for considering the southern hemisphere, the Earth’s rotation is clockwise rather than counterclockwise, which results in the Coriolis effect being reversed (deflection left). Thus, the rotation of high and low-pressure systems in the southern hemisphere is opposite to those of the northern hemisphere.