First of all: As I said above you can NOT think about "rivers of electrons", it is much more complicated than that. You MUST use quantum mechanics, semi-classical models do not work.
I did a quick INSPEC seach and found a review paper from 2003 (L. Collins, IEE Review 2003, pp. 46-49, 2003). This is the explanation used in that paper.
There are actually two effect.
*In unstrained Si the lowest energy level in the conduction band has sex discrete electron states. It only takes a small input of energy from e.g. a acoustic phonon for a electron to change its state. When the lattice is strained the symmetry of the system is broken, the lowest energy level of the conduction band is split, with two of the orignal six levels dropping to a lower energy level and the remaining four rising to a higher level.
This means that it is now much more difficult for electrons to be scattered between the lowest energy states by a phonon, because there are only two states to occupy.
The net effect is that the average distance an electron can travel before being scattered is increased.
*The second effect is that the effective mass of electrons in strained Si is lowered, leading to an increase in conductivity.
I hope that help. It is a pretty good article which is easy to read so if you have access to IEE you can look it up yourself.