I don't know about you but when I am feeling sick, soup. Soup is about the only thing that makes me feel better.
Generally, the symptoms you feel when you catch a cold aren't directly the result of the bug that caused the infection, but rather, a result of your immune system targeting the affected areas (ex. Throat, nose, lungs etc).
For example, you know how your nose gets all runny and you cough up phlegm? That's a result of your immune system reacting to the threat (typically a virus) and does so by kicking mucous production into overdrive. This traps the bug and makes it harder for it to get to its target tissue. A sore throat? That's a result of all the killed throat cells releasing chemicals (called cytokines) that make your throat swollen and tender.
These cytokines are super important in directing an immune system response to the infected tissue. It directs the immune cells that are circulating in your blood stream to the site of infection. It also helps to fight the infection, and as a result makes you feel shitty (one such cytokine, TNF-alpha, is why your body aches when you have the flu.
What the researchers I mentioned found was that certain components of chicken noodle soup inhibit the ability of a specific type of immune cell (neutrophils) to find its way to the site of infection. Neutrophils will kill infected cells, and also make the area a hostile environment (by releasing cytokines) which all ends up making you feel shitty.
By stopping the neutrophils from migrating to the area of infection, it makes you temporarily feel less shitty.
This is all shown In the lab, so it needs further studies to show a real effect in live humans.
Another thing is that the chicken in the soup releases fats that help you feel good and, since the root vegetables are cooked down, they are easier to eat. Which all helps in restoring nutrients and the steam helps loosen mucus, like others have said. Also liquids need to be restored which is easier to do if it taste good instead of plain water. And as it turns out that pure water is HARDER to absorb than water with some salt and carbohydrate in it. So, the ideal re-hydration solution is water with a little salt and sugar added. Add a small amount of protein and fat for calories and you basically have a recipe for chicken soup.