What does the verb 'school' mean in 'a racehorse schools in the paddock'?

The use of the intransitive verb schools is because the author is speaking of the horse in the third person.

The horse is schooling in the paddock. While the horse schools he wears a saddle.

The usage of jargon is not always straightforward in sports terminology. This usage is fairly standard in horse racing.

From an online glossary of horse racing terms presented by Daily Racing Form.

SCHOOLING- Accustoming a horse to starting from the gate and to teach him racing practices. In steeplechasing, more particularly to teach a horse to jump.

PADDOCK- Structure or area where horses are saddled and kept before post time.

Horses are very intelligent animals. But, they often do not enjoy being confined within a starting gate. A large portion of their young training is spent teaching them not to rear up in the gate where they can harm themselves, jockeys, and other track personnel.

Racing practices refers to things like responding to the starting bell, having a jockey and saddle on their back, and not getting spooked by other horses running along side them.

If you ever have the opportunity to take a backstretch tour at a race track, I highly recommend it. There is a lot more to horse racing than just putting the horses in a line and whipping them to go faster!