1. Download the latest realease. It's 0.9.8 as of writing.
2. Install it with their instructions. I ran into a problem though - I have MySQL already but not the files Sphinx needs to install. If you ran into this problem, continue:
3. In console, type:
That will tell you the package you need to install.
4. Mine was libmysqlclient15-dev:
sudo apt-get install libmysqlclient15-dev
5. Try installing Sphinx again, it should work this time.