For Aesop it was all about the animals—the charm of the Fables is the anthropomorphism of foxes, crows and mice. There's little charm to Orwell's menagerie, but he did manage to fit the form of the Greek slave to a modern context. Orwell's great fear was of Communism, not just as an economic theory or social philosophy, but as it was executed in Eastern Europe in all its savage totalitarian oppression. Animal Farm is essentially the history of the Communist Revolution in Russia and its aftermath, but stripped to the essentials so readers can more easily see what was really going on.
The animals initially rebel against the farmer Mr. Jones with an innocent (and true) motto: "All animals are equal." Mr. Jones is a drunk who chronically mistreats his livestock, who in turn revolt in the name of dignity and justice; once the humans are overrun, they set up a provisional communal government purporting to uphold democratic ideals. But no state is run without leaders, and before long the pigs Napolean and Snowball (symbolizing Stalin and Trotsky) assume power. The egalitarian political structure soon breaks down as the pigs introduce more and more qualifications to the original battle-cry. "All animals are equal" becomes "All animals are equal, but some are more equal than others," and a populist government becomes an authoritarian nightmare.
George Orwell was a democratic socialist, so anyone looking for a defense of capitalism will be disappointed. (Not to conflate socialism and communism—a common right-wing mistake—but Orwell wasn't exactly a champion of the free market.) What made him really mad as a democrat (readers of 1984 will know this) was the usurpation of the rights of any population by totalitarian elitists still clinging rhetorically to a false egalitarianism. This injustice is even more horrifying against the supposed democratic ideals initially championed by the Communist revolutionaries; once the leadership abandons the people's cause, the entire enterprise is endangered.
Some readers express confusion at Orwell's use of animals in place of humans. It gives the story a false aura of children's literature, and requires careful analysis to determine each character's symbolism. To both of which criticisms I'm sure he'd say, Certainly. By imagining a slightly irreverent setting Orwell throws readers off guard by the violence and seriousness of his story, to great effect. You can't shock people into recognition or awareness with a steady stream of extreme content, only by introducing it once you've lulled them (much in the way the early Communists were subjugated by Stalin).
As for the second complaint, Orwell was a literary artist first and only after a political commentator and satirist. While 18th century "literary theory" (or what passed for it) objected to any style the meaning of which was not as clear as possible, 20th century critics and authors became obssessed with the concept of interpretation. Of course there were fables and satires before Orwell (Gargantua and Pantagruel, Gulliver's Travels, etc.), but as a 20th century writer his use of symbolism is more an expression of his authorial voice and innovation than an end in itself. (Nor, as in the case of older examples, is it a one-for-one exchange between symbol and meaning.)
On another level altogether, Animal Farm is simply a great book to read, whether you get all the references or not. That's part of its brilliance—readers can still understand and respond to the novel's warning without having to "get" the specifics. And Orwell's elegant style, disarming humor and just plain literary brilliance make it as enjoyable to read as it is instructive, a considerable feat in its own right at a time when most serious literature was overly-didactic and long-winded.