I've been going through some basic forcing and although I'm able to mechanically follow the steps I'm lacking any real intuition for it. In particular I'm confused about the evaluations used to describe elements in $M[G]$.
Suppose $M$ a class/set, $P \in M$ a poset, and $G$ a $M$-generic filter on $P$. Then we recursively define for $\sigma \in M$,
$i_G(\sigma) = \{ i_G(\delta) : \exists p \in G (\delta, p) \in \sigma \}$
Then we say $M[G] = \{i_G(\sigma) : \sigma \in M\}$
Finally we say $\tilde{y} = \{(\tilde{x}, p) : p \in P, x \in y\}$
and $\dot G = \{(\tilde{p}, p) : p \in P\}$
I get that $\dot G$ is defined so that $i_G(\dot G) = G$ and $i_G(\tilde{y}) = y$. But what is the reasoning for $i_G$ in the first place?