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The PS5 needs a specific USB 3.1 external drive formatted in exFAT to recognize expanded storage, and most connection issues come down to either the wrong cable type or the drive not being properly initialized. First thing to try: make sure you're using the back USB port (the rightmost one when looking at the console from the front) since the side ports sometimes have compatibility quirks. The drive itself needs to be between 250GB and 8TB, so if you grabbed something outside that range it won't work. Plug it in, go to Settings > Storage > USB Extended Storage, and see if it shows up. If nothing appears, try a different USB 3.1 cable - a lot of people grab whatever's lying around and some cheap cables just don't handle the data transfer properly.
If it still doesn't show up, your drive might need reformatting. Connect it to a computer instead and make sure it's formatted as exFAT, not NTFS or any other file system. Windows can handle this through Disk Management, Mac through Disk Utility. Once that's done, plug it back into the PS5 and initialize it through that same Storage menu - the console will format it again with its own system, which takes a few minutes. Sometimes a full restart of the PS5 (holding the power button until it shuts down completely, then turning it back on) clears whatever's causing the recognition hiccup.
One more thing worth checking: update your PS5 to the latest firmware version if you haven't already. Older system software occasionally has trouble recognizing certain drives, and a quick update fixes it. If you've tried all this and the drive still won't register, you might have a defective unit, though that's pretty rare.
One thing that trips people up is assuming the drive itself is the problem when it's usually something simpler - I had this exact issue last year and spent way too long troubleshooting before realizing my cable was going bad.
Make sure you're using the proper USB 3.1 cable that came with the drive or a quality replacement, and definitely try a different USB port on the console before formatting anything.
The exFAT requirement is solid advice, but also check that your drive firmware is up to date before plugging it in, since some older external drives need a firmware update to play nice with the PS5's particular standards.
If you've got all that sorted and it still won't show up, the drive itself might be defective - though honestly that's rare.
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