Document FastAPI instrumentation setup

[Harshal96]

Description

Clarifies the FastAPI instrumentation docs for auto-instrumentation environment configuration, trace propagation, WebSocket behavior, and log export. Also renders the opentelemetry-instrumentation README in the generated docs so the opentelemetry-instrument package has a clearer docs menu entry.

Fixes #3301

Type of change

• This change requires a documentation update

How Has This Been Tested?

• python3 -m py_compile instrumentation/opentelemetry-instrumentation-fastapi/src/opentelemetry/instrumentation/fastapi/__init__.py
• uv run --no-project --with docutils --with pygments python3 -c "...FastAPI docstring RST parsed..."
• uv run --no-project --with docutils python3 -c "...opentelemetry-instrumentation README RST parsed..."
• uv run --no-project --with ruff ruff check instrumentation/opentelemetry-instrumentation-fastapi/src/opentelemetry/instrumentation/fastapi/__init__.py
• git diff --check

Full docs build was attempted with uv run sphinx-build -W --keep-going -b html docs docs/_build/html, but this local machine could not complete the workspace sync because the unrelated mysqlclient==2.2.8 build requires pkg-config/MySQL client build flags.

Does This PR Require a Core Repo Change?

• No.

Checklist:

See contributing.md for styleguide, changelog guidelines, and more.

• Followed the style guidelines of this project
• Changelogs have been updated
• Unit tests have been added
• Documentation has been updated

[linux-foundation-easycla]

The committers listed above are authorized under a signed CLA.

• ✅ login: Harshal96 / name: Harshal Parekh (3b7a2e6, 4521300, d1cf86b)

[tammy-baylis-swi]

Thank you @Harshal96 !

Please could you update this reference as per tox -e docs ci/cd:

sphinx.errors.SphinxWarning: /home/runner/work/opentelemetry-python-contrib/opentelemetry-python-contrib/opentelemetry-instrumentation/README.rst:27:'any' reference target not found: opentelemetry_distro

Also could you please add a changelog entry, since it's more than a few lines.

[emdneto]

Thanks for the PR!

Just a heads-up: we no longer update CHANGELOG.md directly. The changelog is now generated from changelog fragments using Towncrier.

Please add the appropriate changelog fragment for this change instead of editing CHANGELOG.md manually. You can find the instructions and expected format in CONTRIBUTING.md.

[Harshal96]

Done!