See example below. I guess the best way to think about it is: it's cheaper to prevent an illness than to treat it. For example, it is widely known that Aspirin helps prevent heart attacks in certain people. Therefor, in a certain group of people it is cheaper to take an Aspirin each day to prevent the heart attack, versus paying to treat the heart attack (ie hospitalization, possible surgery, cardiac rehab, etc).
Avoid crisis rx. The theory of preventative medicine is that the cost of improving health before a condition that is preventable, is cheaper than the crisis to crisis cost of dealing with the problems when they have either appeared or multiplied later in a disease. The extreme example is stopping smoking, which saves money, and by far lessens the disease which almost inevitably will follow chronic smoking.
LESS HOSPITAL/COMPLI. By practicing preventive medicine you prevent doseases and save on the cost of expensive treatment you make early diagnosis and prevent expensive complications and able to cure the condition in early stages there are many examples like early diagnosis colon, cervical and breast cancer preventing flu. Pnuomonia and other infectious diseases and save on expensive treatment and you can go on and on.
Does it? I'll be the devil's advocate here... What do we mean by preventative medicine? Smoking cessation, reducing stress, reducing overeating, exercise more, lose weight - ok that probably lowers costs. Screening mammograms, psas? I'll bet they increase health care costs. More procedures, more treatment, lots of false positives - you get the idea.