One of the big threats of marine debris to marine organisms is ingestion. A lot of the plastic out there is easily mistaken for food, and once consumed creates all types of internal havoc. If they are the lucky, the plastic will pass directly through the animal harmlessly. More likely, it gets lodged in the stomach, signally to the animal that it is full and to stop eating. Pieces can get lodged in a number of places along the digestive tract leading to blockages. Or, if the pieces are sharp they can cut internal organs. All of these ultimately lead to death and probably a lot of suffering.
Seabirds, most famously the Laysan Albatross, are known to eat floating pieces of plastics, regurgitating and feeding the trash to their chicks.
Sea turtles often mistake plastic bags for jelly fish.
Whales, manatees, seals, and most others marine mammals have been documented consuming some form of plastic at some point. The potential for harm is very high. Recently scientists have documented filter feeders & plankton ingesting microplastics. The threat to human health and the implications for the ocean food web is as of yet unknown. But that is for another post.