Well, here's the core of it--spoilers follow, obviously.
The central paradox of the film is that humanity is ultimately redeemed (I use that word deliberately) by the same trait that pushed it to the edge of oblivion: its casual creation and disposal not just of stuff, but of reflections of itself. Our urge to create is a mirror to our urge to procreate: our artifacts are as much our progeny as our children.
WALL-E, left alone amidst the towering remnants of consumerist culture, and programmed with as Sisyphean a task as any factory or office drone, eventually extracts from all that
stuff the spark of what it is to be human. He manages to, from the darkness of mere existence, kindle the light not just of consciousness but of, well, humanity. In doing so, though, he also discovers loneliness.
When he goes to the Axiom out of a desire to protect EVE (and here the symbolism is pretty obvious--not just the name, but the womb-like cavity in which she bears the last natural living thing we know of) WALL-E takes with him what remains of the human experience. He proceeds to, by dint of mere casual contact, to spread that spark to both other robots and what passes for human beings. The marker for the viral meme being spread is the ditty from
Hello Dolly, one of the more mediocre examples of a not-exactly-highbrow art form, the movie musical, which shows just how powerful it is--even in what might be termed a debased and diluted form, it beats the snot out of the void of mere being humanity is in.
The humans on the Axiom, having automated themselves into perpetual infancy, are as isolated as WALL-E was. Bereft of purpose beyond continuing, cut off from each other, they mirror his drone-like directive: they continue to consume and discard, while WALL-E's big brothers below continue to stack and dispose their leavings. The robots on the Axiom have purposefulness at least (until they have breakdowns) but can't go "outside the box" until they bump into WALL-E, as witnessed by MO's deciding to go off the guideline, the button-pusher's learning to wave, and of course the unhinged robots discovering that they can still have a purpose, even if it's not exactly what they were intended for. WALL-E is the snowflake that starts the avalanche.
So with the humans. The captain, nudged by the return of EVE with a living thing from Earth (courtesy of WALL-E) starts asking questions. John and Mary, forced off their guidelines as MO was his by bumping into WALL-E, rediscover the small pleasures of living in reality with other people. They rediscover purpose, physical contact, engagement with their environment--and standing upright. Instead of flailing around waiting to be plopped back into a floating chair, John and Mary are capable of working together on their own volition to catch the falling babies. Instead of taking the word of a long-dead human CEO and a monomaniacal autopilot for it (and IS this really the first time an EVE came back with something, or merely the first time it wasn't swept under the rug by the autopilot?), the Captain decides that maybe they should go check out what's left of Earth for themselves.
And things get better.
That's the story/character analysis. I will leave the more cinematic analysis to a movie person--but even I caught a couple of dozen references, from the obvious (
2001,
Silent Running, any number of Chaplain and Keaton silent films) to the obscure (sight gags from Bugs Bunny vs. Marvin the Martian) to the sublime (I would argue that the film closest in visual style to the first half of the movie is
Lawrence of Arabia).
Italics edit.