Does leading a healthy lifestyle really prevent cancer?

If you understand cancer prevention as taking measures to reduce the risk of developing cancer, then yes, leading a healthy lifestyle will prevent cancer.
But not 100%. It’s always a matter of risk reduction. If you have genetic factors, you may not avoid cancer. Sometimes people just get unlucky: the wrong cell mutates at the wrong time, and you still end up with cancer.
But a healthy lifestyle is part of an overall strategy of cancer prevention, that should include screening as recommended by your doctor, for early detection.
Some concrete examples of a healthy lifestyle and cancer prevention:
Do not smoke

Smoking, the main cause of small cell and non-small cell lung cancer, contributes to 80 percent and 90 percent of lung cancer deaths in women and men, respectively. Men who smoke are 23 times more likely to develop lung cancer. Women are 13 times more likely, compared to never smokers.
Between 2005 and 2010, an average of 130,659 Americans (74,300 men and 56,359 women) died of smoking-attributable lung cancer each year. Exposure to secondhand smoke causes approximately 7,330 lung cancer deaths among nonsmokers every year.
Nonsmokers have a 20 to 30 percent greater chance of developing lung cancer if they are exposed to secondhand smoke at home or work.
Smoking also increases your chance of developing other cancers: oesophagus, larynx, mouth, throat, kidney, bladder, liver, pancreas, stomach, cervix, colon, and rectum, as well as acute myeloid leukaemia.
Eat a healthy diet and maintain a healthy weight.

Obesity increases the risk for at least 13 different types of cancer, see this table.
Obesity is associated with metabolic and endocrine dysfunction, including alterations in sex hormone metabolism, insulin and insulin-like growth factor pathways, and inflammatory pathways. The evidence for a role of sex hormone metabolism and of chronic inflammation in cancer development is strong. Also eating processed meats increases inflammation and cancer risk slightly.
Vaccinate yourself (or your loved ones) against cancer-causing viruses, like Human Papilloma Virus (HPV) or hepatitis B virus (HBV). Vaccination should be a part of a healthy lifestyle.
Be careful with sun exposure. UV rays increase the risk of skin cancers.
Protection against UV radiation can be considered a healthy lifestyle. Avoid tanning beds for sure!