Does not the induced charge on the plane create an electric field?

Yes, of course it does. But don't forget that there is also the electric field of the point charge $q$ above the plane to consider. Remarkably, since this is the electrostatic case, it must be that the electric field of the induced charge precisely cancels the electric field of the point charge $q$ in the region below the plane.

This must be the case since (1) there is no charge below the plane and (2) the electric potential must therefore satisfy Laplace's equation in that region.

Assuming the potential is set to zero at $\pm \infty$, the fact that the grounded conducting plane is, by definition, at zero potential implies that the potential is zero in the region below the plane.

If this were not the case, there would necessarily be a maximum or minimum of the potential in the region below the plane but then the potential would not satisfy Laplace's equation. From [Wolfram Mathworld][1]:

A function $\psi$ which satisfies Laplace's equation is said to be harmonic. A solution to Laplace's equation has the property that the average value over a spherical surface is equal to the value at the center of the sphere (Gauss's harmonic function theorem). Solutions have no local maxima or minima

(emphasis is mine)

Since the potential is constant in the region below the plane, the electric field is necessarily zero. [1]: http://mathworld.wolfram.com/LaplacesEquation.html

Does not the induced charge on the plane create an electric field?

Yes, of course it does. But don't forget that there is also the electric field of the point charge $q$ above the plane to consider. Remarkably, since this is the electrostatic case, it must be that the electric field of the induced charge precisely cancels the electric field of the point charge $q$ in the region below the plane.

This must be the case since (1) there is no charge below the plane and (2) the electric potential must therefore satisfy Laplace's equation in that region.

Assuming the potential is set to zero at $\pm \infty$, the fact that the grounded conducting plane is, by definition, at zero potential implies that the potential is zero in the region below the plane.

If this were not the case, there would necessarily be a maximum or minimum of the potential in the region below the plane but then the potential would not satisfy Laplace's equation. From Wolfram Mathworld:

A function $\psi$ which satisfies Laplace's equation is said to be harmonic. A solution to Laplace's equation has the property that the average value over a spherical surface is equal to the value at the center of the sphere (Gauss's harmonic function theorem). Solutions have no local maxima or minima

(emphasis is mine)

Since the potential is constant in the region below the plane, the electric field is necessarily zero.