Running is not just an effective and efficient way to burn calories and lose weight, tone and strengthen muscles, and a way to get a boost of endorphins, turn a sour mood around, and work out some frustrations and anxiety. Running, like other cardio vascular exercises, is excellent for strengthening your heart and lung health.
Running can be both an anaerobic and aerobic exercise, meaning sometimes it uses oxygen when it has it readily available, and sometimes the body has to perform (run) without oxygen. When you run at a slower pace (i.e. let's say a weekend long, slower run pace), you likely are not gasping for air. In this instance, your body has access to ready to use oxygen. This is opposed to faster sprinting that can be considered a "high intensity" workout, and which the body can only perform for a limited, shorter amount of time because it must do so WITHOUT readily accessible oxygen (and, of course, as humans we can only do so much for a limited amount of time without oxygen). Both anaerobic and aerobic exercises increase one's heart health. In an anaerobic state, the body's preferred source of energy is fat, while it metabolizes and uses carbs during aerobic exercise. Both states help make the heart more efficient, which is what your doctor is implying when he/she says running will help make your heart healthier. This just means that it takes less energy for the heart to pump blood and oxygen through your veins, which lowers the amount of work and stress that it takes for the heart to perform its main duties. As a result, runners often have a much lower resting heart rate than sedentary people, and much lower risks of heart attacks. What's more, a more efficient heart has a stronger capacity to pump blood and oxygen - i.e. when you do perform sprints or anaerobic exercises, and you need your heart to kick in and get moving in order to help sustain you. This occurs when your fitness levels improve, such as what happens over time when you start running regularly.
The good news is that it does not take a tone of miles each week for your body to reap the benefits of the heart health effects of running. Just an hour a week on the trails could mean extreme improvements in your heart efficiency and health.