What does train_on_batch() do in keras model?

It is used when we want to understand and do some custom changes after each batch training.

A more precide use case is with the GANs. You have to update discriminator but during update the GAN network you have to keep the discriminator untrainable. so you first train the discriminator and then train the gan keeping discriminator untrainable. see this for more understanding: https://medium.com/datadriveninvestor/generative-adversarial-network-gan-using-keras-ce1c05cfdfd3

Yes, train_on_batch trains using a single batch only and once.

While fit trains many batches for many epochs. (Each batch causes an update in weights).

The idea of using train_on_batch is probably to do more things yourself between each batch.