Bharat and India are both the same. It's language difference, In english it is called India and in Hindi it is called Bharat. Name "Bharat" is said to come from "King Bharat" who is regarded as our ancestor and India was given by Greeks derived form word 'Indus' which steams from 'Sindhu'. Both name are our pride.
Many people support it because some believes our identity and name should be based on our culture and our language and not given by foreigner in foreign language, which is fair. And some whatsapp university graduates believes India name was given by colonist and we should be called Bharat to de-colonize ourselves.
This is only around social medias. In reality, government has never proposed to change name to Bharat. They just used name Bharat in G20 summit, which is okay given both name India and Bharat are constitutionally valid. This all buzz is created only for Political purpose and trust me all people will forget anything about renaming to Bharat after 2024 elections.