How do I reconnect images stored on external drives with a newly installed Lightroom Classic?

You said "I store all of my images on an external hard drive", but where did you store the catalog? If it was on the same drive, then do what @Rob_Cullen advised and you should be fine. By default the catalog is stored in your Pictures folder on your internal drive, however. If that was the case, then you'll have to retrieve the catalog from a backup of your internal drive. Presets and profiles are also stored on your internal drive by default. https://www.lightroomqueen.com/how-move-lightroom-to-new-computer/

If the catalog and images are in the exact same folder heirarchy on the external drive, AND the Drive appears in File Explorer with the same Drive Letter, then you should be able to simply [Double-Click] the catalog (on the external drive) to re-open it as before.  DO NOT IMPORT FILES!!!

Otherwise, if folders and files appear as missing in the catalog library for some reason, re-linking is required:

MISSING FILES AND FOLDERS-(Computer Darkroom)

MISSING FILES & FOLDERS (Lightroom Queen)