How artist manage to get their paintings exhibited in art galleries?

Do you mean commercial art galleries or museum art galleries?

In the US the artist has to also be a salesperson or hire an agent to represent them. For years, artists only could go around and show their portfolio or send pictures to galleries, today artists can do websites showing their work. It really comes down to good marketing and a lot of luck. Some very good artists never get a break, or do not know how to promote their work, and they do not do as well as lesser talented artists who understand marketing. It also helps to have a lot of confidence in yourself.