Dan Freed discussed in his notes beneath (20.32) the intuition behind the definition of Madsen-Tillmann spectra as "virtual" Thom spectrum, which I reproduce below

I don't think I quite get his idea, like what it means by "virtual" and why this could be more interesting or useful than the real one (I mean the usual Thom prespectrum). Could somebody give an elaborate discussion on his intuition? Of course if you have a different opinion or other intuition please do not hesitate to share with us.