When you learn to play an instrument, you learn how to build connections.
Learning to play the piano builds connections:
Intellectually. Music uses both sides of the brain, and is one of the only activities we know of that does so. Students who take music lessons do better in school (see "Why Piano Lessons?"), and people who play an instrument are less likely to develop Alzheimer's.
Relationally. Music is the thread that ties humanity together; every people group, regardless of time, race, or culture, has music. You may not speak their language, but everyone speaks music.
Emotionally. Learning an instrument allows for creative expression, and is something that can be enjoyed individually (through private lessons), or in a group setting (band, orchestra, etc.). It also requires the student to emote the character of a piece of music, requiring an understanding of emotions and how to express them.
Socially. Learning to play a piece expressively requires communication and translation from music to the audience's ear. It also allows for opportunities for social interaction, from piano recitals to playing for Grandma at Thanksgiving.
Physically. Learning the piano requires dexterity, multi-tasking, and the ability to translate symbols from the page, through the brain, and back out through the fingers. Fine motor skills are required to understand how the body affects the sound of the instrument (the placement of shoulders, wrist, legs, etc.)
Mentally. Music is a language; musicians learn to speak, write, and read this language, and it takes a great deal of memorization, active learning, and critical thinking to translate symbols, and play them accurately and with the correct emotional and musical expression.