I have my piano close to my computer, together with the original music file and a sheet music program.
I use headphones, to listen to the original file, my piano, and the sheet music software "Finale". I use Camtasia to mark and to slow down the music—for some fast notes that's necessary.
If, on YouTube, you find someone playing the piece, it can be helpful to watch the movements of his hands and fingers too.
I play together with the original file, and so I can hear if the notes are right. When the computer plays the Finale file, it is also possible to recognize the mistakes. I hear, into the details, if the music is the same or not. One needs patience, but it's a very interesting activity. I sometimes feel like an archaeologist digging in the sand (my piano) to find the stuff I want. And when I'm finished, I almost know how to play the piece.
At the end, or in between, I print the piece and play it to listen if there are some mistakes left. It's weird that at some places you can feel, rather than hear, that something isn't quite right yet.
I think every music lover has more or less the skill to do it. If you want to try it, I can offer you some lessons to help: contact.