If you haven't already heard the news, Twitter will be discontinuing Periscope this year in 2021. If you have an account on Periscope I recommend backing up your data before it is too late, regardless of how often you use it.
Now it's only a matter of time before your data is ready for downloading. You can close this tab and reopen it when it's ready. Your request will not be cancelled if you close the tab.
For easy access to periscope, you can visit Your Periscope Broadcasts and Your Periscope Data.
If you've done more than watch livestreams on periscope you must repeat the process by going to Your Periscope Broadcasts and clicking 'Send Data Request'. This will prepare the actual contents of your livestreams for downloading.