Here's the passage:
Under the window somebody was singing. Winston peeped out, secure in the protection of the muslin curtain. The June sun was still high in the sky, and in the sun-ﬁlled court below, a monstrous woman, solid as a Norman pillar, with brawny red forearms and a sacking apron strapped about her middle, was stumping to and fro between a washtub and a clothes line, pegging out a series of square white things which Winston recognized as babies’ diapers. Whenever her mouth was not corked with clothes pegs she was singing in a powerful contralto:
It was only an ’opeless fancy.
It passed like an Ipril dye,
But a look an’ a word an’ the dreams they stirred!
They ’ave stolen my ’eart awye!
The tune had been haunting London for weeks past. It was one of countless similar songs published for the beneﬁt of the proles by a sub-section of the Music Department. The words of these songs were composed without any human intervention whatever on an instrument known as a versiﬁcator. But the woman sang so tunefully as to turn the dreadful rubbish into an almost pleasant sound.
He could hear the woman singing and the scrape of her shoes on the ﬂagstones, and the cries of the children in the street, and somewhere in the far distance a faint roar of traﬃc, and yet the room seemed curiously silent, thanks to the absence of a telescreen.
First, the woman is a Prole, and Winston believes if there is any hope that it lies with the Proles. She is their Matriarch, a reminder of his dead mother.
Secondly, he hears the song as he hears the sounds of the city, as if for the first time, because there is no telescreen. He is free, seemingly away from Big Brother's watchful eye.
Third, this scene is synecdoche, a representative part of the entire novel. She sings of April; the novel begins in April, the cruelest month. She sings of "hopeless fancy," and this foreshadows Winston's rebellion: it is a pointless notion. She sing of a "stolen heart," and this is the place where Winston will be caught, where his love for Julia and dreams of freedom and rebellion will all be shattered.
The part where Winston and Julia watch the Prole woman hanging up the laundry is in Part 2, Chapter 10. It happens just before they get arrested by the Thought Police.
Here's what Winston says about why the woman is beautiful. He says it's because her hips, even though they are really wide now, used to be slim just like Julia's. He says that the wide hips are just as good -- why "should the fruit be held inferior to the flower."
I think, though, that she is just a symbol to him. He thinks she's beautiful because she is really still a person. She sings, like birds sing. Party members do not sing because they are not really people anymore. So Winston thinks she's beautiful because she represents to him the hope that maybe there will be real people again in the future.