If you click the icon, top right, with the parallel lines, you can see your history Right clicking any item will allow you to expand it. I dont know how you could save it. But, if you make a screen capture of the expanded folders, you could view it and keep it for later.
The path is
C:\Users\username\AppData\Local\Packages\Microsoft.MicrosoftEdge_8wekyb3d8bbwe\AC\MicrosoftEdge\User\Default\DownloadHistory
But this only leads you to a *.dat file -nothing to view.