The Quake 3 engine is open source. The Quake III: Arena game itself is not free. You must purchase the game (steam, gog) to use the data and play Quake 3 with ioquake3. If you do not want to do that you can play another game made with ioquake3.
Please see the Player’s Guide for help getting started with ioquake3
You can download an ioquake3 test build for