How can I transfer photos from iPhone to SDD with a lightning to USB 3 camera adapter?

Well yes. You need to use an app, like photos app, or the files app to access the SSD and transfer the files to it.

The SSD is not going just appear on the iPhone’s Home Screen.

What did you think was going to happen?

click here ➡️ Connect external storage devices to iPhone - Apple Support

Usually what happens is that the SSD appears in the Files app.

In Photos, you can use the Share menu to Save to Files, and then choose the SSD.

How well all this goes may depend on the iPhone model and the iOS that you are using, although you didn't say.

What app are you using to transfer the files?