This particular case actually went further than the general explanation given below. Killen was tried by two different jurisdictions for two different crimes, despite both being based on the same action, and in one there was no acquittal, meaning that the same jeopardy of freedom was still held over in that case.
The common (mid)understanding lacks a key additional phrase,
one cannot be tried more than once for the same crime by the same authority in a jurisdiction.
A person in a federation jurisdiction who acts in a way to commit similar (or even identical) crimes under both federal and local laws can be tried by both authorities.
If one doesn't succeed,the other is free to take its chance.