Cambridge Dictionaries Online defines "your heart sinks" as an idiom meaning "you feel disappointed or discouraged," and gives this example sentence:
My heart sank when I opened the letter and realized I had not been accepted into graduate school.
"Sink" as a verb means "to move downward, usually through water." For example, the Titanic sank in the ocean. Here, it's being used metaphorically to mean it feels as if your heart is sinking, or moving downward in your body.
It's fairly common for figures of speech referring to negative emotions to refer to body parts metaphorically not being where they should be. Compare, for example, "My heart was in my throat," which means to be scared or anxious.