How do I get into edit mode as shown in this video? I'm trying to follow along with this tutorial, but I seem to have a different version of Blender than is being used in the video, and some parts of the UI react very differently.  At timestamp 9:57, the presenter selects an object with modifiers applied, then presses Tab to enter Edit mode, where he gets a graph of all the vertices and edges overlaid upon the object.
When I press it, I get a completely different view:

I'm using Blender 2.90.  No idea what the tutorial is using, but it's a year and a half old and I can only assume they changed the UI around sometime since this was posted.  I have some minimal experience in Blender, but not enough to figure out what I need to do here to continue to follow the tutorial.  Can anyone explain how to get to the view shown in the video on modern Blender?
You have added the modifiers, but you have not applied them yet. That's why it looks different. The transparent cube on your screenshot is the unmodified mesh. The sphere is the result of the Subdivision Surface modifier. The Displacement modifier is disabled in Edit Mode (see icons of the modifiers).
In Blender 2.9x the apply button of modifiers has been removed. It's now hidden in a drop down menu. Apply the modifiers and it should look like in the video.