"After I cut out the bottom area and try to drag it down"
You know, that you need to Extrude that newly created Face, before pulling it down, right? That's something several people have missed. (Oh, I see that Omar has already mentioned that.)
But maybe it's something else completely, but without screenshot(s) or video, it's hard to tell what's going on.
Sorry video is now public, yeah looking back I think it was because I just pulled down on the gizmo and didn't extrude. Thank you for the help I heard pull down in the tutorial so I thought it meant pull down via grab or gizmo and not extrude my mistake. Haha I look like a fool now than you for the help. Blender is so tough to learn XD
To future people reading this, don't feel ashamed! I started the entire tutorial over again maybe 5 times thinking I had done something wrong here. Nothing I did brought the freshly cut face downward on the Z axis. On run #6, I made CERTAIN I followed every single step and it still wouldn't work....then I saw his ScreenCast Keys show "E" for extrude.
Duh. Obviously. But it just wasn't muscle memory yet so I had completely forgotten. ¯\_(ツ)_/¯
Learned the valuable lesson (for any tutorial really) that if you run into a wall, don't forget to check for exactly what keys they are hitting.