Knowing the dielectric constant of a medium we can earn the electric field at any point in that medium which is deferent from the applied external electric field. So why do we use the Local field correction?
The dielectric constant is a macroscopic quantity - you get this value if you average the effects of many atoms/molecules. If you look closer, you might find small deviations from that average.
You can include local field corrections by working with a dielectric function which includes spatial dispersion. In the case of a homogeneous medium, this means that epsilon is a tensor which depends on frequency omega and wavevector k.
I wrote dielectric function instead of dielectric constant as the effects cannot be included in a single number.