You know from the law of induction that there is a dependence of the field produced by the coil on the applied current. The direction of the field is along the axis of the solenoid, which can be shown by the right hand rule. The Earth's main field is dipolar, with a strong N-S component and a weak E-W component.

In order to find the magnitude of the field, you want to generate a field of equal magnitude in the opposite direction so that your compass spins freely, as if there were no field. In this scenario, you know both that your generating a magnetic field of equal amplitude and orientation as the local ambient magnetic field. If you know the current through the coil then by the proportionality constant you know the field strength.