Is Tamriel Rebuilt free?

Yes. Tamriel Rebuilt is released under Open Source / Community and costs nothing. Neither it nor The Elder Scrolls III: Morrowind has any paid content.

What do I need to run Tamriel Rebuilt?

The Elder Scrolls III: Morrowind, which is itself free. Tamriel Rebuilt is an add-on and will not run standalone.

How do I install Tamriel Rebuilt?

The PlayBound Launcher installs it in one click from The Elder Scrolls III: Morrowind's mods list. You can also install it by hand — full steps are on this page.