D&D 5e lore has precious little to say about the Lady of Pain. A search of DNDBeyond yields only 3 references in the entire 5th Edition corpus.
From the Dungeon Master's Guide:
The city is the domain of the inscrutable Lady of Pain, a being as old as gods and with purposes unknown to even the sages of her city. Is Sigil her prison? Is she the fallen creator of the multiverse? No one knows. Or if they do, they aren’t telling.
From Mordenkainen's Tome of Foes, in a section describing how the Raven Queen is thought to visit the souls of dead adventurers:
The Raven Queen’s reason for communing in this way is a matter of some dispute. Some sages posit that she is using people as pawns in an inscrutable game, the rules of which are known only to her and the Lady of Pain.
From a sidebar quote of a gnome living in Sigil, City of Doors, found in Mordenkainen's Tome of Foes, in the section on the Gnomish pantheon:
Who forged the chains that bind Tiamat in Avernus? Why do the modrons go on the Great March? Who is the Lady of Pain, really? I can’t tell you, but the answers lie in the Golden Hills. And if Garl and his gang don’t know, it can’t be known.
— Griballix, gnome of Sigil
This is every reference to the Lady of Pain in 5th Edition Material.
What resources from earlier editions of Dungeons & Dragons have more detailed exposition on the Lady of Pain?
Best Answer
Several resources refer to the Lady of Pain
...particularly in 2E's Planescape materials, the Planescape Campaign Setting being a good place to start. In The Cage: A Guide To Sigil has mentions as well. The novel Pages of Pain purpots to tell the most about her, though as is frequently noted - even by the book itself - it's full of lies and rumors rather than fact.
Where to get the meaty stuff?
The most instructive appearances of the Lady are in a few adventure modules. Harbinger House depicts one of the Lady's only known limitations and how it comes crumbling down. Faction War gives a bit more of a historical view and depicts a bid to undermine the Lady's authority and seize control of Sigil. Die Vecna Die! has her commanding adventurers on a mission, in her own strange way, and most directly addresses how vast and terrible her power really is.
...but which one explains her?
That would be... none of them. You will never get a canonical answer about the Lady - even the module that comes the closest is full of allusions, suggestions, and hints more than anything direct. Which... is kind of how it's supposed to be. Her role is that of a symbol, an elemental force of awe and wonder in the already awesome and wondrous spectrum of planar weirdness.