quantum mechanics - Action of Fermionic Creation and Annihilation Operators

I'm currently trying to understand the action of fermionic creation and annihilation operators on states in the occupation number representation.

Well, I understand why the bosonic creation operator for example acts by

$$a_i^+ |..., n_i, ...\rangle = |..., n_i+1, ...\rangle$$

on a state given in occupation number representation.

But why is the action of the corresponding fermionic creation operator given by

$$a_i^+ |..., n_i, ...\rangle = (1-n_i) (-1)^{\sum_{j

How can one explain the factor $(-1)^{\sum_{j