Doublethink is essential to the Party maintaining control of Oceania and also to Winston's brainwashing.

Doublethink is holding as true whatever contradictory ideas the Party says are true at any given moment. It also means the ability to instantly switch one's thoughts and beliefs to align with whatever new reality emerges from Party headquarters. 

On a surface level, it's essential because chaos and confusion would ensue if people stopped and questioned what was going on in their world. For example, when the Party announces that the chocolate ration is going up when it is actually going down, if people were to say, "hey, wait a minute, none of this makes sense," it would be much more difficult for the Party to stay in control. Therefore, everyone must be terrorized into acceptance and obedience.

However, the Party's goal, as O'Brien explains to Winston, is far greater than extracting outward conformity and obedience from the people it rules. Its concept of power means forcing people to accept the Party's version of reality as reality. It means, for example, that Winston doesn't merely mouth the words "two plus two equals five" to avoid being tortured while inwardly knowing he is lying. He has to really believe what he is saying before O'Brien is satisfied. This in the end is what doublethink is: wholly giving up any pretense of thinking for oneself. The individual is dehumanized; the state thinks for every person. 

In 1984, doublethink is the practice of holding and accepting two contradictory ideas at the same time. For the Party, doublethink is very important in maintaining control over the people of Oceania. This is because it enables the Party to control information, to rewrite the past, and, more importantly, to make the people believe whatever the Party tells them to.

The importance of doublethink is shown clearly in Part Two, chapter 9, when the Party changes its enemy from Eurasia to Eastasia. Although everybody remembers that Eurasia was the enemy for years, the Party tells them that Eastasia is and always has been the enemy. The only way to accept this is through doublethink, and the people must accept it because if they do not, they risk becoming enemies of the state.

Doublethink also plays a key role in Winston's "re-education" in the Ministry of Love. As O'Brien tells him, doublethink takes manipulation and control one step further: it enables the Party to completely control the minds of the people, creating totally subservient Party members:

Power is in tearing human minds to pieces and putting them together again in new shapes of your own choosing.

I would say that doublethink is necessary for the Party to be able to control Oceania.  Without the ability to do doublethink, the people who run the government would surely go insane and be unable to exert control.

The reason for this is that the people who run the government have to truly believe in what they are doing, even though it is a lie.  If they could not engage in doublethink, they would be unable to really do their jobs because they would not be able to believe in what they were doing.

This is a very difficult concept to grasp -- just look at how Orwell himself puts it

The power of holding two contradictory beliefs in one's mind simultaneously, and accepting both of them....To tell deliberate lies while genuinely believing in them, to forget any fact that has become inconvenient, and then, when it becomes necessary again, to draw it back from oblivion for just so long as it is needed, to deny the existence of objective reality and all the while to take account of the reality which one denies — all this is indispensably necessary. Even in using the word doublethink it is necessary to exercise doublethink. For by using the word one admits that one is tampering with reality; by a fresh act of doublethink one erases this knowledge

As far as Winston's brainwashing, O'Brien is trying to get him to be able to do doublethink -- to accept that 2 and 2 make 5 even though he knows they do not.

What is the importance of doublethink to the Party's control of the citizens of Oceania?

Doublethink is a form of "reality control" in which people accept two ideas, even when these ideas are contradictory.  For the majority of the novel, for instance, Oceania is at war with Eurasia but, suddenly, during Hate Week, the Party declares that it is fighting against Eastasia and always has been.

You will notice from the text that nobody questions this sudden change, even though they can remember the war against Eurasia. This is why the Party is so successful in maintaining control: it manipulates the popular understanding and perception of the past and the people of Oceania willingly accept it because they have been brainwashed to do so. On a practical level, this means that the Party can do as it pleases, safe in the knowledge that nobody will ever question its actions. 

This is, perhaps, best summed up in the following quote from Emmanuel Goldstein from Part Two, Chapter Nine:

"For the secret of rulership is to combine a belief in one’s own infallibility with the power to learn from past mistakes."