> I've never understood them and I'm convinced every definition I've read was written by someone who also didn't understand them
This seems about correct. The book in the list also seems to fall into that category.
Quaternions are actually really simple and pretty once you understand them. The idea is that you specify and axis and an angle. The axis alone gives you a 180° rotation. 1 is the identity and gives you no rotation. If you interpolate between those on a circle (cos/sin) you can get any rotation around that axis.
One may wonder why one needs the sandwich product to rotate a vector when simple multiplication is enough for complex numbers. The reason is that in 2D there is only one plane of rotation and all vectors lie in it. In higher dimensions vectors can be decomposed into a part that lies in the plane of rotation and a part that's orthogonal to it (parallel to the axis in 3D) that is untouched by the rotation. With the sandwich product the quaternions cancel out for the orthogonal part but combine for the part in the plane. This also makes it obvious you have to take half the angle.
Finding the quaternion that rotates between two vectors is also easy. You just multiply them, then take the square root to get the half-angle. Taking the square-root is extremely simple: assuming your quaternion is normalized, you just add 1 and renormalize! Obviously this doesn't work if that quaternion is -1 (can't normalize 0), but it makes sense geometrically: a 360° rotation (-1) is the same around any axis so no axis can be recovered, but for its square root (180°) you have to pick one. It's just not well defined.
Of course all of the above can be shown with a few lines of algebra, and with geometric algebra the whole thing becomes even cooler.