One interpretation is simply that 666 represents humankind in general because of the special significance that the number has in the Bible. Six is known as an "imperfect number" because it is one short of seven, the "perfect number" (seven days in the week, seven tongues of flame, seven spiritual gifts...). So three (the number of the Trinity) sixes is seen as extremely imperfect. Therefore, 666 represents imperfect man, while 777 represents God.
For example, scholars who believe the Book of Revelation refers to real people and events argue that the number represents the value of Nero or Neron Caesar (נרון קסר, Nrwn Qsr*) in Hebrew letters, which also have numerical values