In this problem, we use the Method of Images and get the resulting properties of the charge distribution on the plane. At the end of this process, Griffiths, in Introduction to Electrodynamics, comments that we could have said that the charge on the plane was going to be $-q$ with some hindsight. How can this result have been obvious before arriving at the solution mathematically? I can't think of any thought process that would lead to the same conclusion.
Maybe if I say this less laconically it will be clearer: We are assuming there is no E field below the grounded plane. Look at how your charge density formula was derived. If you don't make an assumption like this you can't figure out the charge density. For instance if the image charge was a real charge there would be field below the plane and the charge on the plane would be zero.
There is no E field below so if we take a Gaussian surface with one side below the plane and the other sides going off to infinity, there is no flux, therefore no net charge, therefore the plane must contain -q.