The full proof is freely available online page 46, Theorem 4.2.6 where Weibel proves that $L_*F$ is a a $\delta$ -functor. For an exact exact sequence $$0 \rightarrow A' \rightarrow A \rightarrow A'' \rightarrow 0,$$ we want to prove the naturality of the connecting homomoprhism $$\partial_i: L_iF(A'') \rightarrow L_{i-1}F(A').$$
Weibel states
I don't understand: what is "the connecting homomoprhism" and how one derives a long exact sequence from the given diagram.
You want to show that $L_*F$ defines a functor from the category of short exact sequences to that of long exact sequences. To do this, you want to prove that if you have a map $t:S\to T$ between exact sequences $S:A''\to A\to A'$ and $T:B''\to B\to B'$, the infinite "ladder diagram" between the LESs of $S$ and $T$ commute.
It is evident they commute everywhere except possibly at the square involving the connecting morphism of $L_*F$. Now Weibel observes that this ladder diagram of LESs is obtained by resolving the morhism $t$, that is, producing a morphism of SECs of projective resolutions, as you have written down, and using the LES on homology after applying $F$.
The connecting morphism in the LESs coming from the SECs of resolutions define that of the derived functors, so to prove that the connecting morphism of $L_*F$ is natural, it suffices to show that the connecting morphism for homology that defines a functor from SECs of complexes to LESs of abelian groups is natural.
This last claim is achieved by a straightforward but perhaps slightly tedious arrow chasing, which I think Weibel does in his book.