How do I keep Screenshot open to take multiple screenshots?

There's also a list of keyboard shortcuts you can utilize:

PrtSc Save a screenshot of the entire screen
Shift + PrtSc Save a screenshot of a specific region
Alt + PrtSc Save a screenshot of the current window

Ctrl + PrtSc Copy the screenshot of the entire screen to the clipboard
Shift + Ctrl + PrtSc Copy the screenshot of a specific region to the clipboard
Ctrl + Alt + PrtSc Copy the screenshot of the current window to the clipboard

It doesn't look like Screenshot can do that, but you can use shutter instead:

sudo apt-get install shutter

UPDATE 2020-08-14: In Ubuntu 20.04 Screenshot can take multiple screenshots.