What motivates people to have cosmetic surgery?

To look better. There is always something that people don't like about their looks. For some people this becomes a source of unhappiness. Cosmetic surgery can change that. They just have to make sure they choose a qualified and board certified plastic surgeon and do their homework to get the improvement that they are looking for.
It Solves Something. Every person is unique, but plastic surgery procedures solve something personal to each patient, whether that be strictly in appearance (lip augmentation), health reasons (panniculectomy), or reconstructive surgery to address an injury or congenital defect. It is human nature to want to look and feel you best and plastic surgery can help you achieve that when other means or methods cannot.
Usually self-esteem. Using breast aug as an example. In a survey of 3000 women who underwent breast augmentation (plast surg nurs. 2008 oct-dec;28 (4):170-4): the majority of women were not at all motivated for surgery to increase their chances of meeting a partner or to please a spouse. Conversely, approximately half of the women were extremely motivated to have breast implant surgery to make themselves feel better.
Improve socially. People with better looks get more social acceptance, they get better jobs, earn more salaries, attract more people to them, feel less depression and have better self image. Imaging your face all burnt with scars or having large tumors in your face or a girl that no boy gets inloved at. Unfortunately human beings have an innate preference for beauty and rejection for ugliness.
Personal. People undergo cosmetic procedures for a range of personal reasons. Improvement in appearance is an obvious reason. Improved self esteem is often a secondary benefit. Most patients desire to look how they feel.
Personal reasons. Some want to look younger, some want to look better, some want to treat a perceived abnormality and so on. The decision is personal to each patient.
New narrative. Beside due to injury or birth defects, it creates a new narrative for oneself Whether it would change the inside is totally different story.
To Be More Confident. In my experience, the reasons seem to come back to a confidence factor. This can stem from wanting to look as good as you feel, wanting to change a problem area that will not go away with diet, exercise and lifestyle changes, to fit into clothing better, or to regain a pre-pregnancy figure. Whatever the reason for surgery, patients want to feel more confident which increases happiness.
Self esteem. Most people are motivated to have plastic surgery to improve their self esteem and confidence that they cannot achieve by other means.
Quality of life. There has actually been a lot of research on the relationship of attractiveness and youthful appearance to success in life: better looking people are happier, earn more, have better relationships, on and on. Even babies gaze at attractive faces longer so it is something hard-wired in the human brain. So while some take it to extremes (vanity), most people have healthy motives.