One great reason for teachers to use that gray space is that it can be a place for assignment directions or resources that you don't necessarily want on your final products.
Open this document for an example: Math Fraction Assignment (Thanks to Google, you do not need to have a Google Account to view this document.)
Usually, the documents we share just contain content on the main workspace. This particular assignment utilizes Google Draw and the gray space to the left.
That gray space isn't very big to begin with. To make it larger, create any shape, like a square. Drag the square to the gray space, off the edge of the screen. As your object goes partially out of sight, notice that your scroll bar at the bottom of the window moves to the right. You can now scroll farther to the left, exposing more gray space.
My favorite use of this space is to use it for math manipulatives. Students can copy/paste or move them to the workspace and use them to demonstrate lesson objectives.
This 'extra space' is available in Google Draw and Presentation. What are some other ways that you would use this space with students?