"Power corrupts and absolute power absolutely." To what extent is this true in Animal Farm?

This is a very accurate phrase to describe the situation which emerges in Animal Farm. Specifically, we can relate this phrase to the pigs and their behavior after the humans are overthrown in chapter 2.
The corrupting influence of power is a major theme in the novel which first emerges when the milk disappears at the end of this chapter. The pigs, who have become the leaders of the rebellion, choose not to share the milk with the other animals. Instead, they take full advantage of their leadership status and keep it for themselves. They then justify it to the other animals by claiming that it has been scientifically proven that pigs are "brainworkers" and need to consume milk for this reason. Power has, therefore, corrupted their sense of equality and fairness.
On the issue of how absolute power corrupts absolutely, take a look at the final chapter of the novel in which the pigs can be seen wearing human clothes, drinking whiskey, and playing cards. In other words, they have been corrupted so absolutely that they have become the very people they once hated.
Moreover, by this point in the novel, they have changed the Seven Commandments into one:

All animals are equal, but some are more equal than others.

This Commandment justifies their every abuse of power and is used by the pigs to solidify their exalted position on the farm. Through a combination of fear and propaganda, the pigs ensure that the other animals are powerless to ever change the situation.