Getting technical about electron velocity
There is a property of solids, called "mobility," used to describe the relationship between the velocity of the moving electrons and the applied electric field. (Most electrons are not moving, they are bound to their host atoms. The electrons which are free to carry electricity are called the "conduction electrons.")
In a vacuum the force on the electron would be qE (where q is the electron charge and E is the electric field--should be a script E with a vector bar on top to differentiate from Energy, but I don't know how to format text on this site) and there is ballistic acceleration just like in a particle accelerator.
In a solid, the electrons do not accelerate ballistically for very long because the they are continually scattering off crystal defects or other things and so they bounce around, moving in the general direction of the E field.
In a superconductor, because of some quantum effects, electrons can pair together (Cooper Pairs) and avoid scattering.
In non-superconducting solids the relationship is
v = µ x E
{That should be the Greek letter "mu" right after the = in case your display font handles optional characters differently than mine} µ stands for "mobility."
There is a limiting AVERAGE electron velocity for a given electric field because the electrons accelerate, then scatter, then accelerate etc. That average velocity in the equation above is called the "drift velocity" because it is just an average net velocity after taking all the scattering into account.
As E increases, the drift velocity increases. However, there is a "saturation velocity" (different for each material) beyond which increasing E doesn't help and mobility doesn't apply. That's one reason why gallium arsenide devices can be faster than silicon devices--GaAs has a higher saturation velocity. Digital GaAs has mostly fallen by the wayside because they haven't been able to shrink device dimensions as well as the silicon industry has. GaAs is still used for some high frequency analog applications. If you shrink a transistor small enough you start to get ballistic effects where some electrons cross the device without scattering, but the new material of this article is meant for the wiring interconnects, not the transistors themselves.
So this new material basically, through quantum effects called "Dirac dispersion," just has a longer mean free path before the electrons scatter. That means a higher drift velocity for the same E field and a higher saturation velocity. (That's why the article says you can use less voltage. Electric field is voltage divided by the distance across which it is applied.) Also important is the fact that this material has a large number of conduction electrons. Total current "I" through a small metal wire on a chip is
I = qnµEA
where q is the charge on an electron
n is the number of conduction electrons per cubic meter
µ is the electron mobility
E is the electric field (given a fixed distance d between ends of the wire, E = V/d)
A is the cross sectional area of the wire
So this material has a large "n" and a large "µ" which leads to a larger current for the same voltage and wire area. But the goal isn't a larger current, which would mean higher power. However, it gives the SAME current at a smaller voltage (lower power) and smaller wire area (higher density of circuits).
I hope this clears things up.