In the end, the reason to choose things in that way is because it works, but in this case it's possible to explain what's going on.
For a warm-up, you can get some intuition for the method by looking at an easier English example of the same phenomenon:
", when preceded by itself in quotation marks, yields a true sentence.", when preceded by itself in quotation marks, yields a true sentence.
To explain the proof you gave, I will use $|n|$ to denote the formula what has Gödel number $n$. In the proof above, $H(n)$ should be read as asserting $F([|n|(n)])$, that is:
$H(n)$ says: $F$ holds of the number of the formula obtained by substituting $n$ into $|n|$
Thus $H([H])$ says,
$F$ holds of the number of the formula obtained by substituting $[H]$ into $|[H]|$
that is,
$F$ holds of the number of the formula obtained by substituting $[H]$ into $H$
So $H([H])$ asserts $F([H([H])])$.
(For informal clarity, I have intensionally not used underlines here.)