Not a Historian but a resident with an interest in local history here.

The site where the Cathedral stands now has had a church standing there since at least 812, possibly longer back to the 5th century. The impetus for starting to build the gothic Cathedral we are now familiar with came when the Archbishop of Cologne in 1164 managed to transfer the bones of the Magi, the three wise kings of the east from the story of christs birth, to Cologne from Milan.

The Cathedral is not the only building in the city that is related to the event by the way , several archways were constructed on the way that the reliquary was carried, i know of at least one still visible today.

Anyway, the presence of the bones lead to a massive influx of pilgrims, Cologne became what we call a "Wallfahrtsort", a holy site that is a destination for a pilgrimage. The old church from the 8th century proved insufficient for the masses, and in 1248 the foundation was laid for the current building. Progress was financed by donations from locals and the revenue that the pilgrim stream created.

Progress was very slow, but not unusually so by the standarts of the time. The first part to be completed was the choir facing the Rhine that houses the reliquary. Afterwards the central cross area. I hope someone else can fill in exact details why progress eventually stopped (I suspect it was loss of fervor because progress literally took decades, + the advent of the reformation putting a damper on church construction funding). But in the the early 16th century work was stopped and the towers were left unfinished with a crane on top.

That crane by the way became a symbol of the city, you will see it on a lot of graphics and pictures from the time, we even have a local saying the the apocalypse will come when the Cathedral is finished.

Construction resumed because Cologne became part of a wealthy new power looking for a prestige project, Prussia. in 1842 Wilhelm Frederick the 4th restarted the project officially. Which was supposed to serve as a symbol of national unity. Important to note that by that point Restoration work and training of a new generation of stoneworkers had gone on since the end of the french occupation in1812. Afterwards it still took 40 years to complete the Cathedral. An important new technology was the use of Steel to build the girder structure for the roof. This prevented the Cathedral from burning down when it was bombed in the second world war.

The building is still not finished today. Attached to it is a place called the "Dombauhütte", a workshop for the Masons that still constantly duplicate and replace parts of the Stonework because of environmental damage. Interestingly the Cathedral used to be white yellowish in colour. The black we are familiar with is the result from it standing next to the central station and decades of coal smoke and environmental polution.