In 1938 Willy Messerschmidt acquired a controlling share in Bayerische Flugzeugwerke, where he worked as chief designer. Anything in production at that time retained the 'Bf' designation, anything entering service after that date became 'Me'. So basically, he took over the business and put his name on it.
If you want the long version, see here:
So which is it? Bf or Me 109?