The easy explanation would be that the shot wasn't taken in 1887, but in 2012. The conversion wasn't from a very old photograph to a new image, but the other way around. It's sort of a "Trompe L'oeil" of photography. Trompe L'oeil (in French) means "fools the eye". That's what's going on here. Rather than be a pretty good restoration of an old photograph, it's the aging of a new image. If you think about it, I'm pretty sure they didn't have chain link fences in 1887.
The first thing that was done was to fade the image by making a copy of the original and changing the blending mode to Screen. That lightened the image without losing a lot of detail. Next a Hue/Saturation Adjustment Layer was used to "tint" the image a slight sepia tone. Another New Layer was made and filled with White. A Layer Mask was added and the Square Marquee Tool (M) used to punch a hole in the center. A huge amount of Feathering (Image/Modify/Feather) was applied to give soft rounded off corners. The Blend Mode was change to Soft Light.
Guides were pulled out from the Rulers to make a loose tic-tac-toe arrangement. A Splatter Brush (B) (drop down the Brush Presets and scroll through until you get to the Splatter Brushes) was modified in the Brush Engine (the painter's pallet looking icon next to the Brush Preset picker symbol. Which is next to the Brush adjustments icon [the hardness and size sliders] on the left of the context aware options bar.) The "roundness" of the Brush was brought down to between 10% and 15%. The Spacing brought out to the point of making the line look fuzzy. On a New Layer, lines were then drawn along the Guides. This New Layer was then put into Free Transform (CTRL T). Right clicking in the selection gives the options for transforming the Layer. Warp was selected and the lines pushed and pulled to give the impression of not so clean a fold. Right clicking in the Layer Box (not on the Thumbnail and not on the name - anywhere else in the box) brings op the Effects Dialog Box. A Bevel and Imboss , an Inner Shadow and an Outer Glow were added to make the "folds" more pronounced.
The Rectangular Marquee Tool (M) was used (on a New Layer filled with White) to make nine squares over the tic-tac-toe arrangement. In turn, the Gradient Tool (G) was used to make Black to Transparent Linear Gradients over each square. The Gradients were loosely drawn on a diagonal from any corner. The idea was to randomize the sheen of each fold.
So there you pretty much have it. Putting an image into "The Wayback Machine" is just an interesting exercise in trying to "fool the eye".