Dear Mylène,
what a wonderful question!
There are some lines of Virginia Woolf which this brings to mind:

"I thought, driving through Richmond last night, something very profound about the synthesis of my being: how only writing composes it: how nothing makes a whole unless I am writing; now I have forgotten what seemed to be so profound."

(from her diaries, quoted in e.g. http://www.nytimes.com/books/first/l/lee-woolf.html)

I find Virginia Woolf’s  phrase very beautiful. I was thinking about it this way for me, playing piano is like feeling whole, feeling at home. It reminds me of the shell the snail carries on his back.

Hi Mylène, all,

Why play music? Well, why listen to music, or more specifically concerts, is discussed here: post .

Why then play. It is clear that it is different to watch a sprinter and to run yourself? To watch Michael Phelps and swim yourself. Similarly with music, the effect is completely different. In all of these cases it develops you. Playing, running, writing, doing things develop people differently than simply hearing, reading and looking at things.

You ask why we take pleasure in playing alone, but then developing yourself is always alone. Writers develop themselves when they write not when they sell books. It is of course different to prepare yourself to play in front of thousands of people, to swim in the Olympics, but essentially I would say - whatever works to push yourself to new heights works.

Playing to other people is also somewhat different than playing alone, as alone your conversation is more with the composer, while when you play to other people the audience takes part in the dialogue. 

Now for the deeper question of what do we develop when we play, I'll leave that one open for now. It is related to the question of why do we speak which as I mentioned in this post has been with me for many years.

Lastly I'll mention Chopin, which I hope to write more about some day, and who I think writes much more to the pianist than to the listener. The experience of playing him is completely different than that of listening to him. What he teaches the pianist is completely different than what he teaches the listener, and it is the more interesting of the two.