I know leading edge vortices over delta wings make more lift. I understood that the core of the vortex sheet is low, so there is suction flow that makes lift. But, I can't understand how a leading edge vortex accelerates the air downward. Please explain the detailed procedure; how does the leading edge vortex make the flow faster?
It is the inclination of the wing which forces the air flowing over it downwards. This is very much like when you hold your stretched-out hand out of a car's window and into the airstream and then slowly rotate it so it has an inclination to the flow. The vortex only makes sure that the air has enough energy to follow the wing's contour instead of separating.
how does the leading edge vortex make the flow faster?
It is suction over the top surface of the wing which accelerates the air and, due to the leading edge sweep, results in a circular motion which is superimposed on the lengthwise motion. Seen this way, it is the low pressure which produces the vortex, and the accelerated, rotating air has less static pressure, in accordance with Bernoulli's law. Trying to separate cause and effect is futile - it is all happening together.