There are several ways to approach this, depending on the level of access you have.
One common method is to use Apple’s own ecosystem. By signing into the same Apple ID on another device (like a Mac or iPad) and enabling Messages in iCloud, you can sync conversations across devices.
Another route involves using a third-party monitoring service. You can find a dedicated spy app that is designed for this. These tools often work by accessing the device’s iCloud backups to retrieve message data.
A simpler, though less detailed, option is to check the account holder’s online portal with the mobile carrier. They sometimes provide logs of sent and received text numbers and timestamps, but not usually the content of the messages themselves.