All the forums on this issue had no solution that worked for me BUT APPLE SUPPORT HELPLINE JUST CLEARED IT UP FOR ME! (I was eligible to talk to them without paying their fee because I was within 90 days of having installed Mavericks (10.9.4) which this forum claimed would fix my problem, but didn't!)
My original problem was that when moving my correctly labelled emails from my 'sent' folder to a new folder that i had created 'on my mac', when transferred, all the mails in the new folder only showed the senders name (i.e. my name).
To solve this issue:
1. Highlight folder that has the issue (I highlighted all the content too, just because...)
2. Go to mail 'Preferences', then 'Viewing'
3. Click 'use classic layout'
4. In the top menu bar, click 'View', 'Column', 'To' (so that 'To' now has a tick beside it)
(a new column appears with the info you want! Then...)
5. Pick up the newly created 'To' column and drag it to the left so it's the first column
6. In the top menu bar, click 'View', 'Column', 'From' (so that 'From' does not have a tick beside it anymore)
7. Back in the preferences box, untick 'use classic layout' and close preferences
8. The problem is solved (and the only thing left to do is drag the extreme right hand border of the Mail window back towards the left so that you can see the body of the highlighted mail/mails)