What does changing "my number" on the iPhone do?

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.

It seems to not do much when you are in the US with a standard phone plan. Whenever I change it, it's only a temporary change. It gets reset back to my real phone number after a minute or so.

The only change it seems to have is, as voidStern said, in the Phone > Contacts app: it just displays your phone number on the top.

If I go into Settings > FaceTime, it shows my original phone number there grayed out, and it says "Verifying". The same exact thing happens in Settings > Messages > Receive At. After a minute, it will no longer say verifying. When this happens, it means that the phone number was reset. You can verify this by going back to the Settings > Phone > My Number setting.

If I enable Airplane mode right after changing the number, it freezes the number change (so it doesn't get reset automatically), but the FaceTime and Messages settings are also frozen on the verifying stage. Even if I enable Wi-Fi the phone number doesn't get reset. Note that after disabling Airplane mode, it took a little longer to reset the phone number, about 3 minutes.

I couldn't find any documentation about it in the iPhone User Guide. However, I did find a few posts on Apple's forms which mention this setting and why you may want to change it. One of them mentions that it changes the number on the Sim card; the other seems to fix a problem when porting over a new number to your carrier.