You’ll get a popup to download the first file, and then another popup to download the second file. Tap that to download both (or either) of the video or audio-only files to your device. Log into your Zoom account, then you’ll see your recording and a Download (2 files) button.
The first link says “Click here to view your recording (can only be used by the host)” – that link opens Safari. Tip: don’t use the second link even though it says “share”! I’ll tell you why later…