How do I create a layer that displays only the bright red areas on the map (The areas of the red layer that are not overlapped by the purple or yellow layer)?
From the screenshot I know you are using ArcMap. To select the bright red area only, you may use the Erase Tool in the toolbox of Arcmap.
This is the helping page: http://desktop.arcgis.com/en/arcmap/10.3/tools/analysis-toolbox/erase.htm
It is a two-step operation.
1. For the in_features, choose the bright red layer, for the erase_features, choose the purple layer, the output layer I would name as bright red layer`.
2. For the in_features, choose the bright red layer`, for the erase_features, choose the yellow layer, the output layer would be what you want to select.