The Hurriyat Conference started out as an organisational alliance of political parties and religious organisations in kashmir aiming at self-determination. In simple terms, they are kashmiri separatists who bestow the idea of an independent kashmir, or its separation from the union of indian states at the very least.
Their Role:- According to the Shimla Agreement, only India and Pakistan Governments are the two parties which have a say in the Kashmir Conflict. But The Hurriyat Conference being an alliance of 26 organisations, must definitely enjoy some sort of support from the people of kashmir (even if only a small faction), and hence cannot be completely ignored. They want kashmir to be declared as a humanitarian crisis which would mean a possible UN intervention and a separate state.
The Indian Government refuses to give The Conference any major importance as Kashmir is still seen as an integral part of india, and considering the struggle for kashmir post independence, even the idea of giving kashmir away doesn't make any sense. Indian Government has had special provisions for kashmir to ensure that it keeps kashmir happy. It recognises the special position that kashmir is in, and it needs to be handled separately. So, India is not the least bit in favour of Hurriyat's ideology and sees them as a danger to india's integrity.