In verbs like "gustar", the "thing" someone likes is the subject of the sentence, and the person who likes it is the indirect object. If you think about it the way Poli explains, the idea is clearer.
- Me molesta el calor.
The heat bothers me.
- ¿Te importa tu salud?
Does your health interest you? (Although we'd rather translate: "Do you care about your health?")
- A María le fascinan las fresas.
Strawberries fascinate María. (But we'd translate: "María loves strawberries.")
- ¿Les parece bien que ponga música?
Is my playing music fine for you? (But I'd rather say: "Would you like me to play some music?")