To set her sights on means she is hoping to go to the Olympics to compete. If you have your sights set on something it generally means you are working hard to get there. You are focusing your time and effort to make it happen.
To match the record means, for example, the youngest person to compete in the past was 14 years and holds the record as the youngest person. If this person is also 14 years old he/she would also hold (have) that same record. Match in this context means equal.