A "fugue" is a complex piece of music in which one or more themes (and variants on those themes) are played or sung simultaneously, in accordance with certain musical conventions. For instance, theme A may appear in the soprano line, while the bass sings an inverted version of theme B; and meanwhile the tenor may be singing a very slow variant on theme C.
In your example, however, "fugue" is used (somewhat loosely) to mean "a constantly reappearing theme or melody in a piece of music". Brian Epstein's presence is likened to that reappearing theme.
("Obbligato" might have been a more accurate simile.)