- " Three girls in their young teens watched the video [Huey Lewis and The News performing "The Power of Love"] along with Marty. They didn't seem to share his enthusiasm. / "Oh, shed that!" one of the girls commented with a yawn. "I only scan that kind of vid at my grandma's!" / "Yeah," the girl next to her added, sounding even more bored by the whole thing than the first. "What do they call it? Rock and rail?" / The third girl shook her head in disbelief. "It doesn't even sound like music!" / "Yeah!" the first girl agreed. "Thank god we didn't have to live in the eighties." She rolled her eyes at the video. "It must have been terrible!" / Shred that? Terrible? Rock and rail? / Marty wondered what they listened to now, but he was afraid he didn't want to know. He suddenly felt very old and out-of-place. He looked away from the video screens and the girls who were too bored to bother. "
- —From Back to the Future Part II by Craig Shaw Gardner (quote, pages 35 and 36)
"Shred" was a term used in 2015, in phrases such as "Shred that". It presumably meant the same thing as "Forget it" or "Let's not and say we did".
This term has absolutely nothing to do with the paper recycling method of shredding.