Isomorphism $\operatorname{Ext}(B,A) \cong H^1(\mathfrak{g}, \operatorname{Hom}(B,A))$

Question

I am giving a talk on Lie Algebra Cohomology, more specifically on the first and second Whitehead Lemma. My problem is that I don't quite understand the proof of the isomorphism $$\operatorname{Ext}(B,A) \cong H^1(\mathfrak{g}, \operatorname{Hom}(B,A))$$ $A,B$ being $\mathfrak{g}$-Modules, which is proved in Hilgert and Neebs Book Structure and Geometry of Lie Groups.
In particular I don't understand why different sections lead to cohomologous cocyles and why this leads to the afore mentioned isomorphism.