The citric acid in the lemon, lime or grapefruit juice makes the flush potion very acidic. When the juice and oil is mixed in a shaker before drinking it produces an emulsion. In fact it is so acid that the stomach doesn't need to add any acid of its own or churn the potion and it passes immediately to the duodenum (Usually within about 15 minutes, approx). This sudden rush of acid and oil emulsion from the stomach causes the liver and gallbladder to supply copious amounts of bile; the flush. The fact that it enters the duodenum as an emulsion assists the flush significantly. Also the pancreas produces large amounts of bicarbonate to assist the bile in neutralising the acid as the intestines is unable to transport acidic material because it would be damaged. In this way the pancreas, liver and gallbladder work together to maximise the effectiveness of the flush.
If only the olive oil was consumed it would sit in the stomach while stomach acid was added and it was then churned to form an course emulsion. The contents would be passed slowly to the duodenum. As far as I know this does not result in a flush.