Retopology method.

I was wondering how you can retopologize a model like this:

The faces he’s creating aren’t clipping trough the mesh and I have the exact same settings he has, but every time I try to do it my way, my mesh always starts clipping it.

Like this:

How can I Retopo like how it is shown in the video?

he’s got ‘XRay’ on.
you can use it in conjunction with ‘Hidden Wire’, and ‘LimitSelectionToVisible’

blender has so many vioprt options that it’s very easy to find a legible mode for retopo.
you should get a pie menu addon that makes all of those accessible, try Wazou Pie Menus.