Proof any CFG can be converted to GNF
The textbook proofs of Greibach NF I have seen, contain endless substitutions. This way to construct GrNF is much crisper. 

The key is to understand what the nonterminal $[Y,A]$ actually represents.
It means a derivation that has a leftward "backbone" which starts from $A$ and ends just before $Y$, as in the picture below.



Now the first production in $H$ can be explained as follows. If I start with $A$ and want to produce GrNF then I must generate a terminal. We choose to generate the leftmost terminal $a$ of the subtree under $A$. So, we produce $a$, but we still must find a derivation $[Y,A]$ of the form in the picture, where $Y$ is the nonterminal that actually produces $a$ using the available production $Y\to a$.