The term "facing" is meant to show the directions of the arrows. It's a bit similar to the Right-Hand Rule, where you align the last four fingers to curl in the direction of the arrow, and the thumb points in the orientation of the triangle. ex: if the arrows are counterclockwise, then the triangle is facing the front since that's where the thumb points.
To check if a mesh is orientable, pick a triangle and assign an orientation and go to its neighbors and assign consistent orientation for them. Repeat this process for all triangles until you find some inconsistency, which implies the mesh is non-orientable. Otherwise, the mesh is orientable.
I've always been bad with using the Right-Hand Rule to visualize things, and this slide is no exception. I understand that the thumb is supposed to point in the direction that the triangles are facing, but I'm not sure about what the purpose of the other two fingers are. The curl explanation that davisgu2012 made makes sense to me, but I was just curious if the orientation of my 2nd and 3rd finger had any more purpose besides figuring out the direction of the arrow.