In Jech's set theory, he says that the properties of the generic extension can be described entirely withing the ground model and also says if we assume the existence of a genric set $G$, $V[G]$ satisfies $\mathbf{A}$, where $V=\{x:x=x\}$ and $\mathbf{A}$ is a formula of set theory, is to be understood to mean "every $p\in P$ forces $\mathbf{A}$".
I really want to konw how the statements about $V[G]$ can be understood as statements in $V$ using the languege of forcing like above. I mean, how are the statements about the generic extension entirely understood (in the ground model) by the forcing language when we even assume the generic set $G$ exists?
And, how can we avoid this assumption to understand statements about $V[G]$ as statements of the forcing language?