In explanations of electrostatic induction of a conductor $B$ by the charge on another conductor $A$, there is often reference to the field lines.
For example, in a capacitor the electrostatic induction is said to be "complete" because a charge $q$ on plate $A$ makes a charge $-q$ appear on the opposite plate $B$.
This is explained saying that "it happens because all the field lines starting from the conductor $A$ with charge $q$ end up on the conductor $B$, where induced charge appear".
Besides making me visualize the phenomenon, I do not understand the physical role of field lines in electrostatic induction: for example, how is the number of field lines starting from $A$ reaching the conductor $B$ related to the amount of charge induced?
Are the fields lines not reaching $B$ completely unrelated to the phenomenon of induction on $B$, or do they play any role?
And, in general, how to interpet/understand the role of field lines here?