How did the protests and demands for linguistic states influence the Central government's actions?

They led to the appointment of the States Reorganisation Commission.

The formation of Andhra state as a separate state inspired similar linguistic movements across the country, advocating for states based on respective languages.
Concerns were raised about the potential risks of separate states leading to division and separatism, threatening national unity.
Under popular pressure, the leadership made a choice in favor of linguistic states, hoping that accepting regional and linguistic claims would reduce the threat of division and promote democracy.
The States Reorganization Commission was established in 1953 to study linguistic and cultural divisions and recommend state boundary changes.
The Commission concluded that state boundaries should align with linguistic boundaries, acknowledging India's linguistic diversity and respecting regional identities.
Based on the Commission's recommendations, the States Reorganization Act was passed in 1956, providing a constitutional basis for the creation of linguistic states.
The implementation of the act led to the formation of 14 new states and 6 union territories, promoting better governance, administration, and representation of diverse linguistic and cultural groups.