You may check FAQ for more detailed explanations about the chemistry.
To make it short I can say that sudden intake of half cup of Olive oil after a day (or more) of fasting gives your gallbladder signal to release a lot of bile (it has been collecting there while you were fasting) to digest it.
Epsom Salts IS magnesium suplhate and it relaxes and opens the ducts making passage of the stones easier.
The strong flow of bile pushes your stones out from ducts into intestines.
Morning magnesium blocks absorption of toxines by your intestines and causes diarrhea which helps to move the released stones out of your body quite quickly. Those stones and other things that get released by your liver during this Flush are very toxic and if they stay long in your intestines you may get poisoned and feel bad and very tired.
So it is importatnt to follow the original recipe. I would not worry about Ornithine though. I never used it and everything went just fine.