My understanding is that the anticommutator relationship shows that the many-particle wave function is asymmetric relative to particle exchange, which is related to the Pauli exclusion principle and also half-integer spin, which, in turn, defines Fermi-Dirac statistic, where the maximal value can't exceed one, unlike in the case of the Bose-Einstein statistics. 

The spin statistic theorem talks about this, you can find it [here][1]. 


  [1]: https://en.wikipedia.org/wiki/Spin%E2%80%93statistics_theorem