Let us consider a point P inside the sphere of earth which is at a radial distance r from center of earth as shown in figure.
Gravitational field at P is sum of the the gravitational field due to the outer shell and the gravitational field due to the solid sphere of radius r.
But according to shell theorem, gravitational field inside any point within the spherical shell is zero.
Hence only the solid sphere is contributing the graviatational field at P. Gravitational field is directly proportional to mass of sphere.
when we move our point p to the centre of earth, the sphere reduced to a point and its mass becomes zero.
Hence gravitational field at centre is zero.