This quote occurs a number of times in the book. In the only times when we know for sure who is saying it, O'Brien is saying it. It is sort of a mantra of the supposed resistance against the Party. Because it is a mantra of resistance, it becomes something of a comfort and a goal to Winston.
Winston hopes, throughout the book, for the day when people will be able to meet in a place where there is no darkness.
He knew what it meant, or thought he knew. The place where there is no darkness was the imagined future...
He takes this to mean a place where there is no longer the sort of control of people that the Party now has -- a place where people can actually let their thoughts and feelings out into the light.
As other educators have noted, this sentence appears in a dream to Winston and is uttered by O'Brien, a fellow party member. When it first appears, in Part One, Chapter Two, it represents Winston's internal desire to rebel against the party's control. But Winston's thoughts on this matter are suddenly interrupted by a "trumpet call" from the telescreen. This interruption forces Winston to abandon his thoughts and, on a deeper level, to realise that resistance against the party is futile.
But, over time, Winston's sense of internal rebellion heightens to a point that he can no longer ignore it. In Part Two, Chapter Eight, for example, he goes to O'Brien's apartment and we hear the phrase repeated again. This is a critical moment for Winston because he has taken his internal rebellion to a new level: he has made himself known to O'Brien and is about to receive a copy of Goldstein's book. The fact that Winston is the one who utters this phrase is indicative of his optimism. Winston thinks that O'Brien has recognised "this allusion" but it is doubtful that he has. It is more likely that he played along to encourage Winston in his anti-party activities.
Ironically, O'Brien is not the man Winston believes him to be; he is a member of the Thought Police and "the place in which there is no darkness," is, in fact, the Ministry of Love, and is a symbol of Winston's torture and reintegration into society. Just like the party's ironic slogans, like War is Peace, irony is at the very heart of 1984 and defines the course of Winston's life.
As indicated in the response above, the quote appeared several times throughout the novel. The very first instance was when Winston recalled a dream he had had seven years back in which a voice of someone seated next to him in a pitch dark room spoke the words to him. Even though Winston was certain that it was O’Brien that spoke to him, he was unsure whether his dream occurred before or after he saw O’Brien for the first time. Winston thought of these words on several occasions and did not know what they meant exactly. He thought that they referred to a future time in which people would live free of control by the Party. Darkness signifies the bondage in terms of thought and action that the people live through under the Party’s control.
The only time these words were actually spoken out was during Winston’s visit to O’Brien’s house. As he bid O’Brien goodbye, Winston said the words to which O’Brien responded with the same phrase as if cognizant of what it meant. Later on while undergoing torture in the hands of O’Brien at the Ministry of Love, it struck Winston that this was the place O’Brien meant they would meet and not a hopeful future as Winston has supposed. At the Ministry of Love, the light in the windowless room were never switched off and they resembled day light.
This quote is important because it represents hope for a better future.