Horses race for several reasons. One they like to show off to each other and racing around allows them to do this. Another reason is to assert who may be the 'best' or strongest. Lastly horses will race because people ask them to.
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Horses will chase anything that the think is going to harm them. Because they are prey animals, they live by the "Flight or Fight" instinct. Most horses will run away from something they think will harm them, but outgoing horses (or often, the herd leader) will attempt to chase away anything that it feels is going to harm it or its herd -- thus giving the other horses a chance to escape.