Anakin needed Palpatine badly, he was the key to saving Anakin's love, Padmé. But, cutting the hand of Windu wasn't the only option to save Palpatine. He could have simply blocked the lightsaber attack with his own lightsaber. Then, he could have tried to make Windu understand the goal of saving Palpatine. It's more natural. There wasn't any risk in that either. Windu wasn't mad enough to kill Anakin for blocking his lightsaber.
Why did Anakin really cut Mace Windu's hand?
IMO it's been a pure (and bad) meta decision (i.e. not really in-character) to support several things:
As one of the comments above mentioned: Yes, it could be to use the element of surprise as well, however to me this doesn't necessarily make sense at all, because at other occassions Jedi have shown to be able to essentially predict movements or things happening (e.g. being shot from the back). So, at least to me, the whole scene always felt at least a bit "cheesy" (same with the whole Order 66).
Mace Windu was going to kill the Sith and that was Anakin's job. The Force made Anakin permanently stop Mace Windu from re-attempting to fulfil Anakin's prophecy hence Anakin's words, "what have I done". Anakin would not have said that if he did not know what he just did. The force gave him the dreams about Padme because the Force realized Anakin was too close to the Sith and too much under the influence of the Sith, but also his love for Padme was way too strong. He was not patient or wise enough to be the chosen one.
Palpatine easily could read Anakin's thoughts, however when Vader was in his suit Palpatine could no longer read his mind. It was completely necessary for Anakin to become suited so that Palpatine could not have sensed Anakin's thoughts, which is why he didnt sense he was going to throw him into the pit while he was electrocuting Luke. Hence at the end of episode 3 when the Emperor asked "are you alright? Vader? Can you hear me?" versus when he was bare skin "save your wife from certain death" reading Anakin's mind like a Dr. Seuss book.
The Jedi were so blinded by their views that they did not realize that they were also unbalancing the Force. Their own rules and lifestyle led to their own downfall. They were so certain that their way of using the Force was the balanced way that they did not assume that in order to bring balance to the force they all had to die as well. The balance was only truly fulfilled after all the Jedi died, and after all the Sith died. Luke and Leia were all that was left, but Yoda and Obi-wan never taught Luke how the council worked or their rules because I think they realized they were also in the wrong.
Not allowing the Jedi to love or have attachment created imbalance and conflict as you can see from Anakin's story. The imbalance this created forced Anakin turn to the dark side. It was simple, all Palpatine had to do was offer Anakin a power he could not go to the council to investigate upon. He had no further knowledge of the information Palpatine gave him because he could not ask the Jedi "can the dark side ever save someone from dying?". Due to their arrogance and lack of empathy, Anakin was forced to either believe this information or not. Because of the dreams he believed it because he was too scared of losing her.
The dark side was a pure force impurity that brought nothing but destruction of millions and selfish needs.
The former Jedi council was also wrong!! Not allowing the Jedi to love or have compassion/attachments caused a minor imbalance in the force. Force-sensitive beings are also humans/creatures of the world capable of all emotions! Denying them of an emotion is denying them of their life in itself.
Not to go off in too much of a rant, the Force guided all of Anakin's actions. We are made to believe he was truly that way, but he was just a force entity the entire time. That is why he became one with the force at the end even though he spent half his life as a Sith Lord, without any training that Yoda, Qui-gon or Obi-wan went through. It was automatic for him. Just like him cutting off Mace's arm, getting chopped up and burned, going into the suit and then killing the Emperor. The force guided everything into place.
In the end all of this is a point of view, so I'm open for debate!
Windu was about to mete out justice in a very un-jedi fashion. But more importantly he was going to kill the key to finding out how he could save Padme. At that moment he made the decision to save Padme at the cost of turning his back on the Jedi Order.
Remember, the Order did not know of his relationship and would have condemned it if they had. The Jedi Order had a strict policy of no personal attachments. Presumably because it would lead to situations just like this where a Jedi would be forced to choose their loved one or the Order and choose poorly(in the eyes of the Order).
It is a cruel irony that his decision most likely lead to the events that would cause her death...
That scene is pretty much the turning point for Anakin. It's the scene that justifies (most of) the rest of episodes 1-3.
5 minutes before this scene, Anakin finds out that Palpatine is the sith lord. He reports Palpatine to the jedi council, but doesn't want to sit and wait. He decides to go back to Palpatine's place (I assume half wanting to see what is going to happen to Palpatine, and half trying to get at least some more information from Palpatine regarding saving Padme).
He walks in the moment Windu and Palpatine are about to start fighting. Palpatine uses the force lighting stuff and Windu directs it back to him, turning him into a grape (I've always thought Palpatine lets himself go rotten-apple-mode on purpose to trigger Anakin's pity).
Anakin sees his only chance to save Padme dying in front of him. He uses the jedi code as an excuse to justify letting Palpatine live. The argument with Windu is still "good" Anakin, using reason to reach his goal. The moment Windu says it's too dangerous to let Palpatine live, Anakin takes the very first real step into the dark side (he had done some things that lead to the dark side, but nothing truly evil yet).
From this point on things escalate very quickly, resulting in the fight between Obi-wan and Anakin at the end of episode 3.
tl;dr : It's a plot device, an event that marks the turning point for Anakin turning to the dark side.
I'm really surprised that no one has pointed this out. I think the explanation is extremely simple.
As Anakin arrives on the scene, this exchange between Windu and Palpatine occurs:
MACE WINDU You are under arrest, My Lord.
PALPATINE Anakin! I told you it would come to this. I was right. The Jedi are taking over.
MACE WlNDU You old fool. The oppression of the Sith will never return. Your plot to regain control of the Republic is over... you have lost...
PALPATINE No! No! You will die!
PALPATINE raises his hands, and lightning bolts shoot out. They are blocked by MACE's lightsaber. PALPATINE is pushed back against the window sill.
PALPATINE He is a traitor, Anakin.
MACE WlNDU He's the traitor. Stop him!
PALPATINE Come to your senses, boy. The Jedi are in revolt. They will betray you, just as they betrayed me.
MACE WlNDU Aarrrrggghhhhh...
PALPATINE You are not one of them, Anakin. Don't let him kill me.
MACE WlNDU Aarrrrggghhhhh...
PALPATINE I am your pathway to power. I have the power to save the one you love. You must choose. You must stop him.
MACE WlNDU Don't listen to him, Anakin.
This is the moment of Anakin's fall to the Dark Side. He is being asked to choose between his loyalty to the Jedi or to abandon them and become a Sith. In this moment, he chooses to side with Sidious against the Jedi. He is not trying to simply stop Windu; Windu is now his enemy. You don't leave your enemy in a position to continue fighting if you can help it. The reason he doesn't kill Windu outright is that he is resisting his choice. That is evidenced when he says, "What have I done?" immediately after Windu's death. Nonetheless, he has chosen to side against the Jedi, and he very quickly accepts this fact fully, as evidenced by slaying Jedi children immediately after.
My feeling was that Anakin lacked confidence in his ability to stop Windu in a sustained confrontation, and he lacked the courage to confront him with words instead of violence.
Simply stopping Windu's blade would have been a more mature choice, the action of a jedi who believed in his power to control the situation. Instead, he acted violently to defeat Windu, without regard for the consequences of his action.
Simply from the movies, the council prided themselves on their ability to deflect bullets, rather than looking to the force for knowledge and defence, they used it for war. They lost sight somewhere what it truly meant to be jedi. They were being fierce warriors and politicians, but not jedi. In OT, you can see Yoda and Obi-Wan became much wiser and powerful because they went back to real jedi ways. Their ability to fight probably went down, but their knowledge of the force was at it's peak. If Anakin was raised this way he would have been a true jedi instead of a warrior. He never looked to the force for knowledge, he was always quick to draw conclusions from sporadic visions he had no control over. This was due to his upbringing. The jedi council should have opened wide arms to Anakin the minute QuiGon presented him to them. They were so sure the sith were gone and that the prophecy would not be fulfilled in Anakin's lifetime. He is the chosen one, and Palpatine knew this because he knew him or his master created him. He knew his purpose, but by befriending him, everything changed. Think ablut it, at the end of ROTS, anakin was gonna kill Palpatine. Had he done that he would have ruled the galaxy and balance would never have beem brought. Or maybe balance had to be brought from the vision of the chosen one. The chosen one gets to decide what balance is or isn't.
It is a difficult question.
I think that it is more a reflex (instinct) action, Palpatine was like a father to him, even if he knows that Palpatine was Sidious, his subconscious only knows that it will save Palpatine and Padme, that is why he said "What I have done?".
But all of this is from my point of view, maybe I am wrong.
