I used to see in the gym people that train with warm clothes while others are with shorts and undershirts. Some of them even put the capuchon on their head while training. What is the reason for those who train with warm cloths? (see an example here and here) it's supposed to be inconvenient and make the body too warm and sweaty.
Keeping your muscles warm reduces the risk of injury. This is, among some other reasons (such as neurological efficiency) why people warm up before doing the actual hard work. If you're doing strength training, there are rest periods between sets of an exercise. Those might be less than two minutes, or substantially longer if you're doing very heavy sets. Keeping warm during that time is useful. In some cases people wear neoprene sleeves over joints (usually knees or elbows) to raise the temperature in those specifically.
For cardio, you might see people do the opposite. Long endurance exercise causes a lot of sweating. The sweat needs to evaporate to keep the body at a comfortable temperature. Since the exercise is continuous you're likely to keep up a good temperature in your muscles and joints anyway. This is why clothes for jogging tend to be airier and lighter.
All of that depends on other factors, though. The temperature of the environment (is the gym kept warm? is there airco?), the weather (humidity) and, perhaps most importantly, personal preference. For some folks hoodies seem to be a fashion statement, or something that lets them focus or stay in the background. It's not always a practical consideration. Some people like to stay very warm, while others prefer to keep cool.
EDIT: I forgot something. Some folks might believe that keeping their body temperature higher will result in more calories being expended and more body fat being utilized. I don't know if there's any truth to that. Intuitively I'd say it sounds doubtful and a case of mistaken cause and effect. People might think it works, because the increase in sweating will result in loss of weight through water. Which, of course, is rapidly regained, especially after replenishing glucose and glycogen stores.