Is Horn of the Abyss free?

Yes. Horn of the Abyss is released under Freeware fan mod and costs nothing. Neither it nor Heroes of Might and Magic III: Complete has any paid content.

What do I need to run Horn of the Abyss?

Heroes of Might and Magic III: Complete, which is itself free. Horn of the Abyss is an add-on and will not run standalone.

How do I install Horn of the Abyss?

The PlayBound Launcher installs it in one click from Heroes of Might and Magic III: Complete's mods list. You can also install it by hand — full steps are on this page.