The sentence is grammatically correct.
Why? It has two short ideas joined by the word and: "The polar bear’s fur is transparent" and "its skin is black." Because each idea can stand alone, using a comma before and is okay. The apostrophe in polar bear’s is correct to show that the fur belongs to the polar bear.
What it means for a 5-year-old: The bear’s fur looks white, but the hairs are actually see-through. Under the fur, the bear’s skin is black.
Simple alternatives you can say aloud:
- A polar bear's fur looks white but is actually see-through, and its skin is black.
- Shorter: Polar bear fur looks white but it's see-through. The skin is black.