Hi Laura
The Story Map apps don't support exposing a layer list/table of contents for web maps. In story maps the model is more that the author chooses the map content and then presents it to the reader as part of their narrative. The exception to this which is in the Story Map Journal where you can create 'main stage actions' - clickable links in the narrative text next to a map that can change the map when clicked on, including turning specific layers on or off.
The Vital Signs app you posted is cool: it's a customization of the Story Map Series app embedded into a web page. For that sort of customization, you would download the source code for the app, work with a dev to make changes and then self-host it.
Another way to get the effect you want: create apps with the Web App Builder in ArcGIS Online that include the layer list widget, and them embed those apps in your Story Map Series instead of using web maps directly. It won't look like the Vital Signs app, but it will enable your readers to choose the layers in the map.
Rupert