As I understand the current languages, Danish, Swedish and Norwegian, they're something like (very) mutated off-shoots of Plattdeutsch.
Why did this language replace Old Norse? Did it happen top-down, because the aristocracy adopted a prestigious foreign language, or bottom-up? Did it happen as an evolution of Old Norse, with more and more words and grammar being replaced, or as a wholly new and distinct language supplanting the old one?
Why did it happen in Denmark, Sweden and Norway, but not on Iceland or the Faroese Islands?
Danish, Swedish, and Norwegian, as well as Faroese and Icelandic, are all just descendents of dialects of Old Norse. They are all of the Northern Germanic branch of the Indo-European languages. Icelandic and Faroese are quite isolated, and have a much more homogeneous population, so they've been more linguistically conservative and closer to Old Norse, unlike Swedish, Danish, and Norwegian.
Edit: replaced Faroese with Norwegian in the last sentence.
Languages are memes (lookup Dawkins' original definition). That is to say that they are socially constructed ideas that change and adapt through time. For every generation they have to be copied and when something is copied there are flaws and these flaws are passed on. In other words Languages evolve, in a sense just like living organisms. The Norse language wasn't abandoned, it just evolved differently in different places.