Rest days from exercise can help one recover and get stronger. The people who are able to safely take few rest days often are those who have been running/exercising hard for a very long time and their body is really use to it. It is so easy to get injured if one does not listen to ones body and take a day off when one need to.
I rarely take a day off from exercise, I alternate between days of running(3 hard miles) and days doing 2-handed overhead dumbbell swing exercises(50-40 lb ones+as many 20 lb ones as I can do: now 17 minutes nonstop). But, I have pushed my limits my whole life and am use to it. These two different exercises I do focus more on different muscles. So I feel like I am having a rest day from one when I do the other. I am glad I do not run every day.
I am very very careful when I exercise and really focus on what I am doing(my technique), so I do not hurt myself and have to take days off. I do regularly stretch carefully(especially before and after doing my nonstop dumbbell swing(cardio) exercise) and do slow easy muscle movements to help certain aching muscles feel better. I find quick and careful squat and back stretches are good to do before and after running or at any time during the day that I feel my legs or back need to be loosened up in order to feel better. I have a strict diet with nutritional supplements. Everyone is different and has to find the dietary and exercise program that works best for them. I also do a lot of meditation and meditative walking that helps me to better tune into my body.
If you push your body too hard in order to try to get in really good shape too quickly you can easily get injured. Often people do not have enough patience to allow their body to become use to handle more of an exercise load. Until one gets into really good shape often rests days are really important in order for one to keep getting stronger at a good rate! Runner often alternate hard days with easy days. But, sometimes a rest day is better for you than an easy day!