Fundamental Rights
In this Part, unless the context otherwise requires, "the State" includes the Government and Parliament of India and the Government and the Legislature of each of the States and all local or other authorities within the territory of India or under the control of the Government of India.
This article just defines a word used throughout the Fundamental Rights chapter. Whenever the Constitution says "the State" in the context of your rights, it doesn't only mean the central government - it means Parliament, every state government and state legislature, and every local authority (like a municipal corporation, panchayat, or police department) that operates under government control. Your Fundamental Rights apply against all of these, not just the Prime Minister's office.
If a municipal corporation official denies you a service because of your caste, or a police officer (a local authority under government control) violates your rights, you can invoke your Fundamental Rights against them - not just against the central or state government directly - because Article 12 defines "the State" broadly enough to cover them.