where does Lightroom Classic store added photos

In reality, LrC stores your images where you choose to have them stored. In the right-hand panel of the import dialog you have full control to choose the hard drive and the folder and whether or not you want the images placed within a date format structured subfolder. But all of that is entirely up to you. Don't leave everything up to Lightroom. Make your own choices. Otherwise Lightroom could put them anywhere, on the local hard drive is probably going to be the first choice.

But I was able to open and edit the photos, with the external drive not attached. The reason this matters is that if Lightroom is putting copies of the photos on the computer's hard drive, I am going to run out of storage very soon.

That means that the photos are indeed on your internal local drive and likely stored in your Pictures directory. To find out where they are: right-click on an image and select "Show in Finder". Be sure to move them from within Lightroom so that they are  moved and not copied instead.

The best way to avoid this in the future is to create and use an import preset. That way if your external drive isn't connected when you import you will get a message saying the external drive can't be found and Pictures will be selected by default which is fair warning to connect your drive and re-establish the destination before you import.

To see if you are working with a smart preview only, check underneath the histogram in the right panel. It will say 'Original Photo', 'Original + Smart Preview' or 'Smart Preview'.

Elie has you pointed in the right direction.

Use the Folders panel on the left side of the Library module to see where your original images are stored.

If LrC is showing a flattened folder structure, right-click and "show parent" a few times as needed to get the true view.

“Add” means “add to catalog only.” But, if you check the box for File Handling > Build Smart Previews, LrC will make low resolution (2560 pixels long side) DNG copies which will be used as Raw surrogates when the external drive is off-line.