To "work like a slave" is a well-known expression for working very hard. What significance does it have in terms of the following quotation from...

Since the animals are compared to slaves in this simile, let’s begin by defining slavery. Vocabulary.com states that “Slavery is the brutal practice of forcing someone to work hard without paying them a fair wage, sometimes without paying them at all.” Sound pretty much like the situation the animals find themselves in?

They are tricked into servitude for the pigs under the notion that after the revolution, the farm will belong to all of them.  Yes, they will need to work to produce food, but it is supposed to be a system of true socialism—with all goods they produce being equally divided up among them. Yet Orwell is demonstrating how difficult is is to maintain true socialism, since someone must always take a leadership role, and it is rare (impossible?) to find someone with the wits and talent to lead others who will not eventually become corrupted by their own power. So it is with the pigs.  

They work the animals harder to make up for the work they and their guard dogs aren’t contributing, to make up for the increasing amounts of extra food and supplies they take, and to make a profit they can spend on the luxuries they wish to buy for themselves. To further make ends meet, they must reduce the working animals’ food rations more and more. Finally, like true slaves, the animals receive almost no "pay" for their labor.

As with many corrupt governments, this process gradually evolves. The animals are all in at first, thinking they are working for themselves, but the pigs are smarter and more adept at manipulation. Soon, without even realizing it, the animals are under the full control of their leaders. Chapter six clearly demonstrates this, following the opening line of, "All that year the animals worked like slaves." We are told that they don't begrudge the hard work because it is to "benefit themselves...not for a pack of idle, thieving human beings.” The irony of this is lost on the animals; they truly can't see that they have replaced Mr. Jones with a pack of idle, thieving pigs. Further dramatic irony occurs when their last day of rest, Sunday, is turned into a work day.  Napoleon tells them it is voluntary; however, anyone choosing not to work loses half of his/her already minimal food rations. Like everything else, the animals simply accept this.

Slavery is a heinous condition, and historically, most slaves have been aware that their circumstances are unfair and inhumane. The citizens of Animal Farm, except for Benjamin, are not even able (or perhaps not willing) to understand that they've been tricked into accepting another form of slavery.  Herein lies Orwell's message: think for yourselves and hold your government accountable for supporting the natural rights and freedoms of all citizens, and never let yourself become a slave to the greedy oppression of another.