How to back up an iPhone to your Mac with Finder

Step by step

Connect the iPhone to the Mac with a cable. Unlock the iPhone and tap Trust, then enter its passcode if asked.
Open a Finder window and select the iPhone under Locations in the sidebar.
In the General tab, choose Back up all of the data on your iPhone to this Mac.
Optionally tick Encrypt local backup and set a password you will remember. See backup passwords.
Click Back Up Now and wait until the progress in the sidebar finishes.

Good to know

The first backup can take a while and needs about as much free space on the Mac as the data on the iPhone.
Each new backup of the same iPhone replaces the previous one. If an older backup matters, for example because messages were deleted since, copy it first.
Backups live in your user Library folder, under Application Support, MobileSync, Backup. Manage Backups in the same Finder window lists them.

Or let TextExhibit do it

Choose the iPhone as the source in TextExhibit and it makes the backup for you over the cable, into a folder on your Mac, then reads the messages from it. It never changes anything on the iPhone.

What to keep in mind

Backups made with Finder are not the same as iCloud backups; both can exist at once.
A backup contains everything the phone backs up, which is private. Keep it on your own Mac and delete it when you no longer need it.

Questions

Why does my iPhone ask for its passcode when I back up?

Recent versions of iOS ask for the passcode before a computer can make a backup, to protect your data. Enter it on the iPhone.

Should I encrypt the backup?

Encryption also saves passwords, Health and other sensitive data, and protects the backup with a password. If you turn it on, you will need that password to read the messages.

Can I back up over Wi-Fi instead?

Finder can sync over Wi-Fi once set up with a cable, but a cable is faster and more reliable for a full backup.