How does the use of Leading Edge Root Extension help in increasing stall angle?
The idea is to generate a vortex near the fuselage. In most cases we do not want any vortexes, as the unnecessary air movement always causes an increase in drag. In this case however, you need them.

As the air flow separates from this small triangle wing while at large angles of attack, it gains a rotary motion (if you look at the left wing standing in front of the aircraft it will rotate clock-wise). This vortex inducts further swirling movement across the wing. 

Now, if you are familiar with fluid dynamics, you can imagine that this will decrease air pressure at the top of the wing (not much impact on the bottom side), which is what we want - higher lift is generated. 

What is more important in this the case, is the boundary layer. If you provide it with energy, making it more turbulent, it will pay off with sticking to the wing surface even while flying at large angles of attack. Obviously the larger is the angle without stall, the higher lift can be generated. Therefore an airplane with wing extensions can perform sick maneuvers like cobra and other more useful in aerial combat. Of course it requires a huge amount of spare thrust, but you do not have to worry about that in military jets.