Why is my Red-Eared Slider turtle suddenly not eating well and how do I know if it's trying to hybernate?

Well, your turtle would only be trying to hibernate in the winter. So it's not that. You should take him to an exotic vet or reptile specialist. If your turtle is a female and being housed with a male then she is probably pregnant. Pregnant turtles tend to eat less. But you should still take him to a vet.