An enormous structure like that has several benefits for a potential future civilization.
For a highly egalitarian/eco-friendly civilization (Like Star Trek), the taller you make a structure, the less impact it has on the surrounding wildlife. With everyone concentrated into a small area, no one is chopping down forests, draining wetlands, damming rivers, and carving up mountains to make space.
For authoritarian governments, concentrating people and industry into a smaller area provides easier control, and in the event of a revolt, easier destruction.
In simple economic terms (For industrial buildings and such), the closer together the facilities for extraction, processing, refining, and manufacturing the more efficient the manufacturing process is.
There may also be applications which require enormous amounts of power or resources, such as a defense laser, wormhole generator, or planetary shield. Depending on technology levels, these things could require enormous reactors, heatsinks, and support machinery.
Whether or not colonization would render arcology irrelevant depends on a couple of things. Food is heavy and inefficient to transport. It would require enormous amounts of interstellar infrastructure to effectively transport a meaningful amount of food back home. In a universe where interstellar travel is cheap (due to space elevators, fusion rockets, probably some form of ftl travel), this is feasible. It is more likely though, that colonies will not only take hundreds of years to get the kind of agricultural base to support such a large population, but also won;t be able to effectively transport resources back home.