Why doesn't Messages recognize my contacts? Ok, so this summer I was in Korea. I used a Korean SIM card in my iPhone 4S while I was there. I ended up losing my iPhone before I came back to the US. 

I recently purchased an iPhone 5 and backed up all my stuff that used to be on my old phone. Everything works fine except when I text (for the first time) new contacts that I have recently added. 

When I do this, their number shows up in the Messages App instead of their name even though I have their number and contact info saved in my Contacts. When I click on "add to contact list" in the Messages App and I choose the contact, it adds a +82 in front of the number. 82 is the country code for Korea. I don't know why it does this. Can someone please tell me how to fix it?
Go to Settings > Phone

Check that My Number is has been updated to your new number.

At times when changing SIM cards, the number does not update in this area.  This will cause issues like you have experienced.  I recently changed carriers (in the USA from AT&amp;T to T-Mobile) and experienced a similar problem after swapping the SIM card in an unlocked iPhone.

See Apple Support for more information regarding swapping SIM cards.