It helps to understand the difference between the two theories. First of all, these are theories based on collected evidence, and represent scientists "best guess" as to how modern man got where he is today, in terms of global distribution. Evidence to support both theories abounds, so let's stick with basic differences here.
The multiregional hypothesis contends that modern man developed globally on different continents, from different locations, roughly around the Pleistocene era, about 2 million years ago. Evidence is used to support the development of modern humans from an ancestor called "Archaic Homo Sapiens", the predecessor to modern Homo Sapiens, or modern man. Outside of normal migratory patterns between the continents, the multiregional hypothesis basically says man evolved globally and did not start out on one continent, then travel to all the others.
The Out of Africa hypothesis contends a much shorter time-table, 100,000 to 200,000 years ago, and says there was a single origination point for the evolution of modern man, that being the continent of Africa. Evidence again, is offered to substantiate the evolution from common ancestors, then details a migratory pattern originating in Africa, spreading through Europe, Russia, across to the North American continent, down through the South American continent.