I Just Woke Up To Find My Male, Red-eared Slider With Bits Of Broken Shell Floating In His Tank. What Could Have Happened?

Shells of the Red-eared Slider or any other turtles can be broken due to injury. These shells are used as protective sheet for skin. Damage/broken shell can expose the skin to may pathogens like bacteria and fungus to cause infection. You should examine the turtle very carefully and note any injury to the skin.
Now many vets leave the broken shell area without any repair work to heal naturally. Only infection or injury is treated. But many vets repair the broken parts of the shell to avoid infections.
In my opinion your local vet can help you about the treatment plan if any. Your local pet stores can give some information which you can share with the vet.

Turtles shells do shed occasionally. I had one that suddenly started losing bits of his shell but he was perfectly healthy. What does the shell underneath look like? If the shell, where the pieces came from, seems hard and normal it's just a natural shedding process. If it's soft or revealing skin then I would seek veterinary help immediately. It's probably nothing to worry about unless it starts coming off in big chunks, and it's very unlikely that it was caused by the other slider. Sliders don't just get violent for no reason and I've never known them to turn on each other like that.
 
My advice is, unless his skin is exposed, just watch him, and if he starts losing more shell or acting unusual take him to the vet.
 
Good luck and I hope your turtle is okay :)