How does Newton's corpuscular theory explain the speeding up of corpuscles when entering a denser medium? I can't find an explanation for this anywhere. Intuition would imply that the corpuscle would slow down. I mean a person running at a constant speed enters a crowd of people or a forest. The presence of obstacles would cause a reduction in velocity. How did Newton think it would speed up the particles ? 
 A: In order to explain the refraction of light in going from air into a medium Newton assumed that the medium applied an attractive force on the corpuscles which was perpendicular to the surface.
This force accelerated the corpuscles as they entered the medium in a direction which was perpendicular to the surface of the medium whereas the component of velocity parallel to the medium was unchanged.   
 
So for Newton his intuition was that if there was this attractive force then the corpuscles would speed on entering a medium and from that assumption Snell's law could be derived with the refractive index being the ratio of the speed in the medium relative to the speed in the air.
Newton had no explanation for the speeding up other than if he made that assumption he could use his principles of mechanics to derive Snell's law.
For him the good news was that using his mechanics with some assumptions Newton could also derive the laws of reflection.
Your intuition is perhaps biased by a whole body of knowledge regarding the atomic theory of matter and how the wave theory of light correctly predicts the slowing down of light as it enters a medium.
A: They were talking classically in analogy to sound wave. Sound wave travel faster in metal.
See this post. Did Newton argue that particles speed up when entering a more dense medium? and http://www.physics-and-radio-electronics.com/blog/corpuscular-theory-light/ under the title "Newton’s Corpuscular Theory Statement". 
I found another article which made more sense see http://galileo.phys.virginia.edu/classes/609.ral5q.fall04/LecturePDF/L20-LIGHTII.pdf page 3 which explain it with conservation of energy.
It's almost just an physical assumption. In his age, the usual physical sense is different, he probably picked the knowledge in mechanic wave from sound because the theory of EMM doesn't exists unitl hundreds years later. Bascially, he coulnd't do the math, because, remember he just invented calculus, which was not yet wildly accepted.  
