Why is a charge induced on a grounded metallic plate when another charge is placed near it? Why doesn't the charge flow into the ground?

When we bring a charge -Q close to a plate some positive charge is induced on the near surface of the plate to this external charge and some negative charge on the other farther surface. I believe this happens because of the electric field produced by -Q.

But when the plate is grounded why don't electrons flow from/into the ground and neutralise the induced charge?

One might say that charge flowed from the grounded into the plate to make its potential zero. How does it make the potential zero?

Starting from Maxwell's equations we can derive Poisson's equation for electrostatics

$$\nabla\cdot\vec{E}=\frac{\rho}{\epsilon}$$

That is, the gradient of the electric field is proportional to the charge density at each point in space.

This means that in a neutral material, the gradient of the electric field is 0, and therefore the electric field is constant.

So if the charge flowed so as to make the conductive material neutral, then the electric field coming from the nearby charge couldn't terminate and would extend into the conductive material. This current would flow toward the surface facing the nearby charge. But at the boundary of the plate, the charge has to stop flowing because the material on the other side of the boundary is nonconductive.

So what happens to the current? It flows just long enough to build up a surface charge that 1. terminates the field lines from the nearby charge, and 2. Repels charge from within the plate so that no further current flows.

why don't electrons flow from/into the ground and neutralise the induced charge?

In a sense they do, because with a nearby external charge of -Q, the total charge produced on the surface of the plate is +Q, and therefore the system as a whole is neutral.

One might say that charge flowed from the grounded into the plate to make its potential zero. How does it make the potential zero?

It makes it zero because we chose to use the ground as the reference point for measurement of potential, therefore we define it to be zero.

The negative charge $$-Q$$ induces charges on the initially neutral plate.
$$+q$$ on the side nearest the $$-Q$$ and $$-q$$ on the other side with the plate still neutral.

The plate is earthed with the $$-q$$ charges moving "as far away as they can" from the $$-Q$$ charge leaving the plate with a net positive charge of $$+q$$.
This could be check by first removing the earth connection and then the charge $$-Q$$.

In terms of electric field lines, one could say that for the $$+q$$ charges to move, the field lines which were originally going to the $$-Q$$ charge from the $$+q$$ charge must be replaced by other field lines, but there is no other source of such field lines.
Thus the $$+q$$ charges are "locked" in position whereas the $$-q$$ are free move.