The water in the Kern River is cold so it's a good idea to wear a wet suit like Caroline is wearing.
And for safety, one should wear a helmet and a floatation (life) jacket in case you fall into the river.
Also note that Caroline is wearing river shorts over her wetsuit. It may look funny but it's purpose if to help prevent you from slipping off the raft when it's wet, as well as reduce wear-and-tear on the wetsuit.
Clothing/Shoes/Weather Gear: It's a good idea to wear a bathing suit or T-shirt underneath your wetsuit. That way, if it gets too hot in your wetsuit, you can peel down the top half without violating any indecency laws.