The FBI's argument is that there could be information on the phone that could help stop future terrorist attacks.
Specifically, the FBI is asking for a "a signed software update which will disable an iPhone feature which deletes data after a certain number of failed attempts at guessing the PIN (which, along with a per-device secret, is the seed from which the encryption key is derived). On iPhones with relatively short PINs, this effectively "breaks" the encryption because a small key space can be quickly searched." They can then use this insecure operating system to break into the phone.
The FBI is not concerned with this compromising the security of other devices, because in order for someone to use this back-door they would have to gain access to the file that only the FBI and apple have access to. The FBI trusts its own security measures, and apple users already trust apple not to let their private key for system updates fall into the hands of hackers. So from the FBI's perspective, there isn't a substantial risk to apple users and there is a significant possibility of preventing future terrorist attacks.