The Internet is right - running as a means for weight loss is an excellent option, because it revs up the body's metabolism and burns calories. To lose weight, the science is relatively simple: the body must be consuming and taking in less calories than it is expending. When you are running or exercising in general, you are helping really burn extra calories. Coupling exercise (calorie expenditure) with healthy and moderate eating (decreased consumption) will help you lost weight.
The science behind how the body burns fat, and specifically how it burns fat through running, is a bit more involved though. You have probably heard of the "carb load" before a long run or a big race. This is referring to when runners eat a particularly carb heavy meal, such as a heaping bowl of pasta, a large side of rice and sweet potatoes, or a delicious stack of waffles or pancakes. The idea is to take in enough carbs to fill the body's glycogen tanks up. Glycogen is the body's first option for "fuel" when it is in an anaerobic state, meaning it can't readily access oxygen because it is working so hard (i.e. sprinting, weight lifting, high intensity exercises, etc).
The body burns FAT, though, when it is burning calories in an aerobic state. This is because oxygen is needed (i.e. aerobic) to oxidize fat. Slower, longer distance running and walking, swimming, biking, and other endurance sports will burn fat more readily. Essentially, if you are focused on just burning fat, focus on lower intensity physical activities, longer and slower forms of exercises, and maintaining optimum fitness levels (as fitter individuals with more muscle mass tend to oxidize fat more efficiently).
However, it is important to keep in mind the old saying, "you can't out exercise a bad diet." If you want to burn fat (or carbs or really make any dent in burning enough calories to lose weight), you will need to make sure your food and nutrition is on point. You should focus on getting in a lot of high quality protein (animal products or legumes and beans if you are a vegetarian or vegan), as well as nutrient rich sources of fats and carbs (oils, avocados, and dairy are great quality fat options and whole grains for carbs). And try to keep excessive sugar intake to a minimum, saving those for special occasions.