Courtesy of Yahoo Answers, accuracy not guaranteed:
This is a kind of amalgam of two phrases originating in this transition period.
The first one was used to precede some strong assertion
and would have translated as something like 'By my Faith..' (meaning, by my Christian religion) .
The second part ('Begorrah') probably originated as follows: being reluctant to say 'By God..' , the common people said something like 'by Gor' , pronouncing it 'Be Gor' , which in time became 'Begorrah'.