The LTTE was set up by the Indian intelligence agency RAW kinda like how the Pakistani ISI helped set up the Taliban. They provided training, arms, air dropped food and medical supplies, etc. However, in the beginning it was only one of many Tamil separatist groups. Later on when the Indians decided to become an honest broker in Sri Lanka they sent in peacekeepers with a mandate to disarm all the tamil groups. The one that resisted was the LTTE which ended up killing 1100 Indian soldiers. It was India's vietnam. Gandhi, however, refused to relent. It was his successor who removed the troops. When it looked Gandhi was going to be re-elected, the LTTE feared he would again crack-down on the LTTE (not just in Sri Lanka but in India and other surrounding countries) so they assassinated him.
There is of course depth to the story. Indian troops did commit a few atrocities of their own while embroiled in the conflict. And the LTTE did and still continues to find significant support in the Indian state of Tamil Nadu (where most of the world's tamils live). Indeed, RAW trained and ran most of its LTTE support from TN. However, Gandhi's death essentially ended whatever sympathy the wider Indian population and the Indian defence and security establishment had for the LTTE. India remains concerned about achieving what they call a 'just peace' with significant rights for Tamils. But the establishment sees no need for the LTTE and would love to see them destroyed. And that's become a sore spot for the Tamils. Even the ones here routinely complaint about the lack of Indian support…yet few of them know the history.
