It's not "game engines" that do this - all the 3D software you use does this. It just doesn't tell you about it, and your professor seems fairly under-qualified if he doesn't know this. They exist in the computer's memory even if the software is hiding them from you. All 3D programs have an option that will make the triangles visible. They will also have an option that splits them into editable edges, so you can play with them. But they were always there to begin with and it's naive of your proffessor to be teaching this subject and still wondering "what triangles are for".
A triangle is the only way to arrange verts and guarantee a flat surface. When you have a quad, you can arrange the verts in a way that it must bend. But it's already made of triangles and it's those triangles that allow the bend.
