He makes it look easy! That was also one of the first tutorials I watched.
To extrude the circle, select it (or the arc that remains, in his example) and use the Extrude tool. In his case, he used the mouse on the blue arrowhead to move it exactly in the proper axis. Note that you can have the youtube player slow down the playback! You can follow his exact movement at ¼ speed, I would think.
In the next sentence, he adds two edge cuts. I know the Ctrl-R tool does this, but I’ve not gotten it to do two at once symmetric from the center like he did — various modifiers to control that, and the mouse wheel to change the number of cuts. Adding one at a time accomplishes the same thing.
When he did that, the faces bounded by the new edges were selected already. If that’s not the case, go to face-select mode and select just those faces (and only those). Now use the Scale tool (press S or use the toolbar) and it will change the radius of that section of wall but not affect the ends (which are not selected).
You might try a basic tools tutorial first, and then come back to this. He makes extensive use of Inset and Scale, selecting entire loops at once, and adding loop cuts.