From the song, by Paul McCartney.
He believes in the possibilities of love and happiness akin to that which he knew in the past, even though neither of those states are currently part of his life.
“He,” of course, being the speaker, not Paul McCartney. McCartney may well believe something entirely different.
Scrambled eggs.
Of course, this was the Sixties, so it didn’t really have to make sense…
He was so confident the song would be a hit, he couldn’t wait for the song to finish to say so. The Beatles infused several songs with this type of meta-commentary—other examples include “She Loves You”: Yeah, yeah, yeah! and George’s characteristically downplayed “Here Comes the Sun”: It’s all right.
It sounds nice, though it doesn’t mean anything, and it makes a good song to sing when you’re drunk and maudlin with a bunch of friends.
Oh, baby how I love your legs.
“I believe in yesterday” = “I wish things were like they were before.”
This. And that he still believes in their love.
It means Paul McCartney never has to work another day in his life.
And on a similar vein, what one thing did he say that was “wrong” enough to end his relationship?
'Cause like saying one thing wrong isn’t going to end a relationship. Unless it was something like:
“Shut your festering gob, you tit! Your type really makes me puke, you vacuous, coffee-nosed, malodorous, pervert!”
Then, perhaps.
Or “your sister is better in bed”. I can think of any number of "one thing"s that could be said to end a relationship. 
Stupid cut and paste. Yes, I know it’s supposed to be “toffee-nosed.”
Serves me right for believing everything I read in teh intrawebs.
Well let’s say what we’ve got:
*"Why she had to go
I don’t know
She wouldn’t say.
I said: “Shut your festering gob, you tit! Your type really makes me puke, you vacuous, coffee-nosed, malodorous, pervert!”
Now I long for yesterday."*
A bit of a stretch fitting it into the meter of the song but certainly plausible.
Sir Paul does owe us an explanation.
Well you know, Love has a nasty habit of disappearing overnight.
“I am the Walrus?”
More specifically, the yellow matter custard dripping from a dead dog’s eye. She was eating at the time, and it represented just how unconsiderately oblivious he could be.
I’m pretty sure I read a biography some years ago that indicated the song was written after a beloved relative passed away. Perhaps I’m thinking of a different song.
There’s a lot of detail in various Beatles/McCartney books about the melody but surprisingly little concerning the lyric. For example, in the McCartney biography Many Years From Now, author Barry Miles has only this to say:
'It has been suggested that the lyrics are about the loss of Paul’s mother, and one line* could possibly be read as that. If so, it was an unconscious element in the song’s composition.*
The rumour goes that when Paul’s mother died he cracked an off-colour joke about her which, in due course, he came to regret.
There’s nothing about the lyric in Revolution in the Head - The Beatles’ Records and the Sixties (Ian MacDonald) or in The Beatles Anthology or other literature I recall reading. However Wikipedia notes as follows, presumably from one of the sources it lists, but I don’t know which:
Also, in John Lennon’s opinion extracted from The Beatles Interview Database:
** The one line in question must be ‘I said something wrong’. Speaking of which, in Eva Cassidy’s version of Yesterday she sings ‘I said some things wrong’, thus giving more probable cause for her guy breaking up with her in that he perceived her misdemeanours to be plural.
Songwriters have been, are, and always will be asked about the meaning of their lyrics. One such, whose name currently escapes me, responded to his questioner with something like ‘Man, it’s all I can do to get the words to rhyme!’
I may have read it in “A Twist of Lennon”, by Cynthia Twist, John’s first wife.
I disagree with Lennon’s take on the song. It’s a lament for a happier past in the singer’s life and that’s all. It doesn’t need any more resolution.