What was the significance of the announcement made by the British just before Independence?

The Princely States would become legally independent.

Prior to Independence, the British announced that with the end of their rule, the paramountcy of the British crown over the Princely States would also end.
This meant that the Princely States, numbering 565, would become legally independent and could choose to join India, and Pakistan, or remain independent.
The decision of joining India or Pakistan was left to the princely rulers, rather than being determined by the people.
This posed a significant challenge as it could potentially lead to the fragmentation of India and threatened the unity of the newly formed nation.
Problems arose immediately, with the ruler of Travancore announcing their decision for independence, followed by the Nizam of Hyderabad.
The Nawab of Bhopal was reluctant to join the Constituent Assembly.
This situation created the possibility of further divisions and the formation of multiple small countries within India.