iPhone doesn’t have any native way to save or export your SMS messages to a computer. This can be done using third party tools.

Option 1 – Paid software

Use purpose build, commercial software, such as:

• www.iphonesmsexport.info
  
• www.iphonesmsexport.com
  

Instructions can be found on developers websites.

Option 2 – Do it yourself

You can extract messgaes from iPhone iTunes backup. This is what most paid software does anyway. Doing this yourself involves few more steps, but is not too complicated. Here are exact step by step instructions:

Extract SMS messages from iTunes backup:

1. Connect your iPhone to iTunes and perform a backup.
2. Navigate to iPhone backup folder on your PC. Depending on Operating System it can be following locations:
   • Windows 7 / Vista: C:\Users\<user name>\AppData\Roaming\Apple Computer\MobileSync\Backup
   • Windows XP: C:\Documents and Settings\<user name>\Application Data\Apple Computer\MobileSync\Backup
   • MAC OS X: User\Library\Application Support\Mobile Sync\Backup
     
3. You will see one of more folder with long ID name, such as d34fde66ba898c63212ef6640ef94073b00a346c. If there is more than one folder, go inside one with the latest Date Modified.
4. You fill see large number of files long long ID numbers. Find one with name 3d0d7e5fb2ce288813306e4d4636395e047a3d28 and copy it to your desktop.
5. This is where all your SMS messages are saved. File is SQLite3 database. If you open it with a notepad you can actually see all your messages, however they won’t be in an easily readable format.

Export messages from SQLite database:

Again, there are two ways of doing this. Both are free this time:

1. Go to www.iphone-sms.com upload your 3d0d7e5fb2ce288813306e4d4636395e047a3d28 file and download exported data in Excel, PDF or HTML format. This is easiest way to get data exported and it is presented in nicely readable format.
2. If this doesn’t work or you worried about security of uploading your SMS file, you can extract data yourself, although it won’t be in such a nice format:
   • Download sqlite3.exe. More info and latest versions here.
   • create folder sms on your C: drive, and move sqlite3.exe and 3d0d7e5fb2ce288813306e4d4636395e047a3d28 files there.
   • rename file 3d0d7e5fb2ce288813306e4d4636395e047a3d28 to sms
   • open command prompt and run following commands (one by one):
     cd c:\sms
     sqlite3.exe sms
     .output exported.txt
     select * from message;
     .exit
   • All messages will be exported to exported.txt file.

Tested on:
Windows 7 PC
iPhone 5 with iOS 5.1.1