Delta Keyframes, how to use If I want to keyframe a simple location change then I just set a key frame by moving the object to its initial location, press I, select location keyframe, move the timeline cursor to when the change ends, move object to next position, make another keyframe as before. The animation runs as expected.

If I do the same using delta keyframes FROM THE I DROP DOWN, I was expecting the object to move the same as above relative to whatever starting position I put the object in. But the object doesn't move. In the Properties, Object tab the delta transform numbers remain at zero. The object stays at the end of the cycle.

I know how to duplicate the object and type in numbers into the Object tab Delta transforms to produce the same movement but offset. I also know how to convert [all of the] keyframes to delta keyframes. I just do not understand what the purpose of delta keyframes in the I menu are for and how they work.
Delta transforms are used when you have some incorrect location in your animation, like if your animated object is at wrong place you can just adjust the location in Delta Transform.
If you don't use Delta transform for such repairing, you have to edit a particular keyframe by overwriting them.
For example, if you have a plane on which a cube is floating from one place to another, you will obviously make a Location keyframe by pressing I. But the problem arises when you want the same animation for another. Dupicate the animated cube at another place, and play the animation, you can't see the another cube in animation, you can see only one cube. Did it disappear?
No, it is just on the location where the original cube is lying during the animation.It has the same Location path which the original cube has, as it is been duplicated!
To fix this, you should use Delta Transform, assume that you duplicated the cube on X axis, now the path should move parallel to the original cube i.e on Y axis, but not on same X axis, so change the location of X axis in Delta Transform, and you'll get the parallel movement of both cubes i.e without affecting your animation movement.
What you will do in absence of Delta Transform?
As I mentioned above, you have to overwrite each keyframe by changing its location again and again by pressing I and selecting Location keyframe, depending upon the number of keyframes you have created. But this makes it easy if you want to change the initial location of any axis and make it permanent till the last keyframe. It will have the same animation but on different axis, which you have changed in Delta Transform.

Also Refer this.