How can I better understand that det(A) > 0 => rotation and det(A) < 0 => reflection?
@pavelkang: Take a look at the slide on determinants, and see if you can figure out the answer to your question. (If you do figure it out, post the answer here!)
I think the idea is determinants tell how the volume changes. For reflection, the orientation is changed so the area becomes -1 * original area. That's why the sign of the determinant distinguishes reflection from rotation.
Right: the geometric interpretation of the sign of the determinant is that it tells you whether orientation was preserved (+) or not (-). Two orientations, and two signs.
This relationship holds in any dimension, though one must be careful to recognize that certain operations that are orientation-preserving in an even number of dimensions are orientation-reversing in an odd number of dimensions. Strange fact of life. :-) See my comments on the determinants slide for more.
The listed conditions are necessary and sufficient?
Yes; the rightmost column gives the defining property for each type of transformation.