What does changing "my number" on the iPhone do? I noticed that there is a setting under Phone > My Number which displays my phone number. What does changing this setting do, and why would someone want to do that?

Is there any Apple documentation that explains the feature?
It changes your number everywhere on your iPhone (That is, it doesn't change your number on the SIM or the network). 

An example where your number is used is under contacts in your phone app. If you scroll to the very top it displays the number you've set in the preferences. So really, it doesn't change much.