There is no way to download the official data. However, there are some websites that offer this functionality.

One website I found is here. This lets you see 3-day history for free but up to a year if you pay and appears like it would on the device.

Alternatively, on some stories, you can click share story at the bottom, copy it and send it to yourself.

There is no official way to access the data. However, there are some websites that offer similar functionality.

One website I found is Sensor Tower. It lets you see 3 day history for free, and up to a year if you pay. The feed appears as it would on an iOS device.

Alternatively, for some stories, you can tap on the Share Story button shown at bottom, copy the link and send it to yourself. However, this approach is highly manual, would have required following it since the day App Store redesign was publicly launched. Also, it is not possible to do so if you want to start out today as the older feed entries can no longer be accessed.