The first verse of the Beatles' version is:
They say that ev'ryoneWants someoneSo how come no oneWants me
There's a contrast even just in the words since "ev'ryone," "someone," and "no one" all rhyme with each other and leave "me" by itself, but this is illustrated in the melody too. "Ev'ryone," "someone," and "no one" are all sung to higher pitches (B A B, B A G, and B A, respectively), but "me" is sung to a comparatively lower pitch (G). Musically, then, there's a sense of "me" being apart and excluded.