What does he that falls in love with himself has no rivals mean?

Oh, dude, that's like saying if you're so into yourself, you won't see anyone else as competition because you're too busy admiring your own reflection. It's basically a fancy way of saying Narcissism makes you blind to other people's awesomeness. So, like, maybe take a break from staring at yourself in the mirror and notice the cool folks around you, you know?