Carlisle had been a vampire for a few hundred years old before he met Esme. When she was 16 years old, she had fell from a tree, breaking her leg in the process. Carlisle, who had been working in the same area, treated Esme.
Several years later, when Esme lost her unborn baby, she tried to kill herself by jumping off a cliff. Doing so didn't kill her, but locals thought she was. When Esme was brought to the morgue (still alive) Carlisle heard her heart beating, albeit faintly. He decided to turn her into a vampire, thus making them mates.