This quote by Mozart is wonderful.
I think we often focus upon the notes on a page of music and adjusting their duration, volume and tone to achieve a desired sound. Rests however, can be easily missed along the way. It is important to always incorporate all the indicated rests in a piece of music, for this is what allows the music to breathe! 

It is not only what me make of the notes which shape the music, but also what we make of the silence. The sound of silence can be anything from mysterious, to gloomy, to surprising. When you come across silence and rests in your music, think about their function and purpose. Why did the composer put a rest in that spot, rather than sustaining the notes beforehand? Is it to lighten the texture of the music? Is it to create drama? If you can understand the placement of silence in a piece of music as well as the story that the notes tell, you will have developed a great interpretation of the piece.