1. The measured dark energy density is half a joule per cubic kilometer, if I remember correctly. Why does this cause a negative pressure?
For easy numbers, suppose the density was 1 joule per cubic meter (much higher!).
Imagine a cylinder with sliding piston, that contains 1 cubic meter of space. For simplicity suppose the cylinder cross-section is one square meter. Take this device outside the universe for study.
Carefully pull the piston out one meter, until the volume in the cylinder is 2 cubic meters. There are now 2 joules of energy inside. You must have done one joule of work in pulling the piston outwards (or else what made the extra joule?). Therefore you exerted one newton of force for a distance of one meter.
The pressure in the cylinder is minus one pascal (-1 newton per square meter).
The calculation is easy and in fact shows that if the Lambda energy density is X joules per cubic meter then the pressure
it exerts is minus the same number of pascals: -X newtons per square meter.
This proportionality (called w, or the "equation of state") is -1 in this simplest case but it is something theorists play around with to see how the model depends on it.
The other question is
(2) Why does negative pressure promote expansion?
This is a consequence of the Friedmann equation---the basic equation used in cosmology. It has a pressure term which for roughly 80 years was largely ignored because stars and galaxies and dust do not exert significant pressure.
Pressure couples to gravity, somewhat as energy does. So positive pressure (like a concentration of mass-energy) can cause collapse---it exerts gravitational attraction just as mass does.
However negative pressure does just the opposite!
This is explicitly spelled out in the Friedmann equations, which I will save for another post.