Learn Turkish: Tavşan ve kaplumbağa ormanda yarışırmış. - The hare and the tortoise would race in the forest.

This is the inferential aorist tense, which combines the aorist suffix -ır (used for habits or general truths) with the inferential suffix -mış (used for things you heard or stories). Together, they describe a habit in the past that you didn't witness yourself. It is heavily used in fairy tales and fables to mean 'they used to' or 'they would'.

What is the difference between saying yarışırmış and yarışırdı here?

Both translate to 'they would race' or 'they used to race.' However, yarışırdı is used for past habits you know as a fact or witnessed yourself. Because this sentence is about a famous fable, we use yarışırmış to show it is a story that was told to us.

The subject is 'the hare and the tortoise' (they). Why isn't the verb plural, like yarışırmışlar?

In Turkish, when a plural subject consists of animals, plants, or inanimate objects, the verb usually stays in the singular form. Since a hare and a tortoise are animals, yarışırmış is grammatically correct without the plural suffix -lar.

Is yarışmak only used for physical racing?

The verb yarışmak means to race, but it is also the general word for 'to compete' in any context. You might also recognize the root noun form, yarış, which simply means a race or a competition.