To see the new feature, make sure your Shazam app is updated, open it, and swipe left-to-right to go to My Shazam. This section shows your most recent Shazam and the new discovery date. If you scroll down to the "Recently Shazamed" section and tap "See All," the app provides the date and timestamp for every song that you found through it over the past few years.
If you scrub on the right side of the screen, you can quickly jump to any specific month over the past few years to see what you were listening to at that time. Tapping on these songs brings up their main page, which have links so you can jump into Apple Music and add them to your library.