According to Martin Stig Andersen, sound designer and composer for Inside, the use of a human skull was essential in in the production of the game's soundtrack. In a Gamasutra article, which I just happened upon, Andersen explains the idea behind the method of using a skull to distort the synthetic music in the game.

"Early on, as we were working on Inside, I had the idea of working with a human skull because I think it's very interesting how the sound of your own voice sounds very different in your own head."

Andersen has even uploaded "pre-skull" and "post-skull" samples of the game's music. I really recommend scrolling through the short article for interesting tidbits, like how the teeth of the skull eventually fell off one by one from the vibrations of the music. Pictures of the skull are of course included in the article.

@dudeglove said:

One word: Interns

I mean, it's interesting in theory, but considering the result doesn't really sound like something you couldn't accomplish with any reverberating material, it seems more like he experimented to see if it would sound like something inside your head and stuck with it because he wanted to record with a skull.