http://sethares.engr.wisc.edu/paperspdf/consonance.pdf
Informal overview:
http://sethares.engr.wisc.edu/consemi.html
This is almost certainly an oversimplification (e.g. the listener's familiarity with the sounds also matters), but the results match my own perception of consonance and dissonance reasonably accurately. It doesn't address the question of "why", but intuitively it seems that dissonance is associated with sounds that take more mental effort to understand. If the partials line up closely then they are perceived as part of the same sound, and if they are spaced far apart then they are obviously different, but there's an awkward intermediate spacing that's uncertain (the "roughness" on the Plomp and Levelt curve). Dissonant sounds contain a lot of this uncertainty.