Complex sentence: That same year she became the first American singer to win the first prize at the International Music Competition in Geneva, which raised Rankin's profile in the opera world; she soon received invitations to perform at many of the world's best opera houses.
Split into simpler sentences: That same year she became the first American singer to win the first prize at the International Music Competition in Geneva. This competition win raised Rankin's profile within the opera world and she soon received invitations to perform at many of the world's best opera houses.