In this video retopology is done with addon b-surface using annotation to create mesh, https://youtu.be/FkWvgT75SbA?t=793 at 13:13 he moves vertices while having Annotate tool active. Bottom left corner says the key pressed is EVT_TWEAK_R but i don't know what it is. I am using the default keymap of blender and the tips on bottom show call menu for me instead of select and move like in the video.
They use right-click for selection in the video. It's a preference setting: Preferences... → Keymap → Preferences section → Select with Mouse Button left / right.
With right-click for selection you can use the Annotation tool with the left mouse button, and drag vertices around with the right mouse button. In Selection mode, new vertices can be extruded with Ctrl + left-click. The context menu is opened with the W key.