Docs feature: auto-build documentation

### What do you see as an issue?

It currently takes a few minutes each time to build the docs. So, each small mistake requires a few minutes to wait for the docs to re-build.

### Solving the problem

I tried using [sphinx-autobuild](https://github.com/executablebooks/sphinx-autobuild), but I haven't been able to get it to work.

### Anything else

I'd be happy to work on a PR with someone who knows restructured text/sphinx better than I do.

### Are you willing to submit PR?

- [ ] Yes I am willing to submit a PR!

### Code of Conduct

- [X] I agree to follow this project's [Code of Conduct](https://github.com/apache/airflow/blob/main/CODE_OF_CONDUCT.md)

Start by reviewing the repository's documentation build setup and the linked sphinx-autobuild project. Determine how the docs are currently built and define a workflow that rebuilds them automatically after source changes; done means small documentation edits can be checked without manually restarting a full build.

Written by the indexing model from the issue text.