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.

PS: the answer given by Gregory below explains the motivating for introducing virtual Thom spectra. It would be even better that somebody can elaborate more on the motivation of using negative-dimensional vector bundles to define Madsen-Tillman spectra and the specific usage of negative dimension in it.

Actually in Freed's notes there is one comment right after this which says that $MT$ does not only stands for "Madsen-Tillmann" but also for "tangential variant" of the Thom spectrum i.e. the Madsen-Tillman spectra are tangential and unstable. I wonder whether this "tangential" and "unstable" are also tied to the negative dimensionality.