How do I transfer iPhone 13 photos and videos to an external drive?

The "best" way to do this, is by first copying the photos/video to a computer, and then, copy/move them to an external drive.

However, if you really don’t have/ want a computer, this is trickier and more finicky. In this case, you'll need:

• A Lightning-to-USB 3 Camera Adapter, plus
• A USB flash drive/SSD (often needs power via a Lightning power pass-through or powered hub),
• Then use the Files app on iPhone to move items from Photos → Save to Files and then copy to the external drive.

You're very welcome! Since you wanted to leave iCloud out of the equation, you are left with only a few choices. Transferring files from the iPhone directly to an external drive isn't really that complicated, but does come at an additional cost for the adapter, and potentially, a powered hub ... not worth it if you only need to do this seldomly.

As an alternative to using the native Files App, it is possible to use a third-party file manager App to backup or move data to a variety of external storage options - including third-party Cloud services, external media devices and network storage. Of those available from the App Store, one App that comes readily to mind is FileBrowser Professional by Stratospherix:

https://apps.apple.com/gb/app/filebrowser-professional/id854618029

You would be well advised to review the extensive online Manual and help files to determine suitability for your needs.

https://www.stratospherix.com/products/filebrowserprofessional/filebrowser-professional-user-guide.php

Speaking of personal experience, I am yet to encounter a file storage or file/photo backup scenario that the FileBrowser App won’t handle. I routinely use USB and third-party Cloud storage for this purpose. FileBrowser also integrates directly with the native Files App, further simplifying file/photo management. There is a learning curve, but it works.

Your iPhone 13 has a Lightning port. A simple and cost effective method of connecting USB storage to your iPhone will be to use an Apple Lightning to USB3 Camera Adapter:

• Lightning to USB 3 Camera Adapter

https://store.apple.com/xc/product/MK0W2ZM/A

This adapter has both a USB Type-A port to connect your USB storage device - and, essentially, a Lightning pass-through power connection with which to connect a Power Adapter. Connection to external power is often required as your iPhone alone may be unable to source sufficient power for the connected USB storage.