Let's start by defining our terms because complementary and alternative medicine (CAM) as a recognized approach to medical care is a comparatively new concept. Complementary and alternative medicine has three categories and is defined by the National Institutes of Health's (NIH's) National Center for Complementary and Alternative Medicine (NCCAM) as a variety of different health care "systems, practices, and products" that are different from and used with or instead of conventional (i.e., allopathic or Western) medical systems, practices or products:
CAM: "a group of diverse medical and health care systems, practices, and products that are not generally considered part of conventional medicine." (NCCAM)
The three categories comprising CAM are complementary medicine, alternative medicine, and integrative medicine.
Complementary medicine is CAM used along with conventional treatment, for example, acupuncture or homeopathy together with conventional pain management treatments. Alternative medicine is CAM used instead of conventional treatment, for example, acupuncture used instead of allopathic pain management treatment.
Integrative medicine is more refined than the first two and combines conventional medicine with well proven CAM systems of treatment, having "high-quality evidence of safety and effectiveness," in a single integrated treatment; it deliberately employs both CAM and conventional medicines.
To address your question, there are probably as many ways that CAM helps our bodies as there are CAM options. Yoga is certainly a CAM practice that is used by many as a complementary medical approach, though it is not noted as being used as an alternative medicine (see definitions). Yoga helps bodies by adding to balance and flexibility and by increasing stability in vital signs such as blood pressure and respiration (though not noted as a replacement for conventional therapies for conditions related to blood pressure and respiration).
Acupuncture, according to Chinese Medicine, helps bodies because it clears a body's energy pathways (meridians) and, in so doing, eliminates pain pathways, relieves symptoms in organs, revitalizes vitality and mental processes. Often used as a complementary medicine, some use acupuncture as an alternative medicine in many kinds of cases.
Integrative medicine (see definition) helps bodies by giving the best of both approaches, the conventional approach and the alternative approach. For example, if Indian Ayurvedic medicine treatment can be integrated with allopathic treatment, the body can be revitalized and balanced and nourished at the same time that harsh prescriptions with unkind side effects are being administered. The outcome will be more efficacious results and a gentler recuperation and a lessened rebalancing of homeostatic systems.