As Andy says, your veteran status is determined by the number of previous Battlefield games you've played that you link to your account. To become a veteran, follow the steps listed on this site:
You need to link your Gamertags, PSN IDs and old EA master accounts to your current EA account for the Veteran's Program to have the most information possible to find your Battlefield games. You can do the steps below to link your accounts.
Step 1: Link your Xbox Gamertags, PSN IDs and old master accounts to your main EA account that you will use for the Veteran Program.
Start linking your accounts here.
Step 2: After linking your accounts, login to the Veteran's site with your EA account to auto-check for your games. If a game isn't found, you may be able to manually enter the details to add it.
Veteran Rank: You will receive Veteran rewards in select Battlefield games if you have a Veteran rank of 1 or higher. You need two full Battlefield games to be rank 1.
The Veteran's site can be found here.
By becoming a veteran, you will get a special in-game rank and unlock the M1 Garand.