Why did Eli Whitney's cotton gin completely energize cotton growing?
Eli Whitney's invention energized the cotton industry because it allowed cotton to be processed much more quickly than it ever had been before. Before it can be used, cotton must have its fibers picked off of its seeds. Before Whitney's invention, the the sort of short-staple cotton grown in the South could only be removed by hand in a process that took one slave a whole day to clean 1 pound of fiber from 3 pounds of seeds. This made the cotton more expensive to use.
Whitney's machine was able to do this work 50 times faster than it could be done by hand. This meant that short-staple cotton (which could be grown in more places than the long-staple cotton) was now more economically viable. With that increase in productivity, the cotton industry in the South was energized.
This question illustrates what happens when something which has been done by hand is now capable of having a machine do the work. The cotton industry had been very labor intensive, requiring many hands to harvest a crop, clean it, bale it, deliver it to the mill etc. The cotton industry was one of the biggest users of slave labor because so many people were required. When the cotton gin was invented, the requirements for labor dropped to almost nothing, and cotton became a profitable crop again. Where profit is possible, people follow, so the cotton industry came alive again with many more people planting cotton.