The did it in 2 stages. Before that, you need to know some history. A fire burnt down the majority of downtown. So, they had to rebuild everything anyway. The downtown was also low enough that the sewer would flow into the bay at high tide. After the fire, they made certain parts of the downtown streets up to 2 stories higher to accommodate a functional sewer system.

First, they made the streets higher without touching the storefronts. "Crossing the road" would have required climbing ladders. Once the streets were in place, they made the sidewalk and the new storefronts on the old first or second story. So, when you walk around that part of Seattle, there is sometimes just two stories of air beneath the sidewalk. You can identify these sidewalks through their use of amethyst blocks which actually serve/served as sunlight. As you can imagine, a block can consist of a single large structure and so sometimes the basement of these buildings are huge empty spaces. If I recall correctly, the Seattle underground is actually disappearing as development occurs and put these large basements into use. However, I do believe there is a movement to try to preserve at least part of the underground by simply buying enough adjoining buildings and funding their upkeep with tours (which is where I got this information). In terms of touristy things, I think definitely one of the more unique things in the USA and probably has similar vibes to the catacombs of Paris (without the bones).