Robin's fate and storyline in Batman v Superman was not really touched upon in the film — except for the defaced Robin costume hanging in the Batcave (which fans saw pretty early on in the trailer). Several fans believed that Robin's backstory might link to The Joker's origin — some even saying that Jared Leto's Joker in the DC Cinematic Universe was at some point the second Robin — Jason Todd (Dick Grayson being the first as per DC Comics).
Now director Zack Snyder has shed some light on the mystery of Robin's fate. "In my mind, it was that Robin had died 10 years earlier, during some run-in with a young Joker," Snyder told IGN. "So there was a fun backstory there to play with. I felt like the whole idea was that there had been loss and there had been sacrifice. In a weird way, he had sacrificed everything to be Batman. He doesn't really have a life outside the cave. I thought that including Robin, a dead Robin, was helping us understand that he's been on quite a little journey."
While Snyder did not say which Robin it was that died, or reference any other story as influence, it does sound pretty close to how Jason Todd met his fate in the DC Comic story "A Death in the Family", with a similar effect on Batman's state of mind.
In an earlier interview, Ben Affleck spoke about how Robin's death had led to Bruce Wayne/ Batman becoming more bitter and angry. "He's bitterly disappointed in the past that he's lost this guy who fought by his side," said Affleck talking about Robin (via Entertainment Weekly). "That character's death must have been devastating to him, and he's suffered. We get the sense that he's suffered a lot of devastating losses before this movie even starts."