I am having trouble figuring out how an integration works.

Consider the function, y = x^{2} . The antiderivative is x^{3} / 3 or in other words its x^{2} (x/3). How come multiplying the original function by (x / n) (where n represents the exponent + 1) will lead to the area under the curve.

Put in another way, let's define the above in terms of a graph. Consider the function y = x^{2} and you want to find the area from 0,1.

All you are doing to find the area is x^{2} (x/3) . You are creating a rectangle in which the dimensions are x^{2} by (x/3). If the function we were dealing with were x^{3}, the rectangle that would yield the area under the curve would be x^{3} by ( x / 4)

How come the above rectangle just happened to be the total area under the curve? How did this work out and why does it work out?

In order to try and figure this out, I searched the history of calculus and found this from wiki, but I didn't really understand it:

Spoiler

In this paper, Newton determined the area under a curve by first calculating a momentary rate of change and then extrapolating the total area. He began by reasoning about an indefinitely small triangle whose area is a function of x and y. He then reasoned that the infinitesimal increase in the abscissa will create a new formula where x = x + o (importantly, o is the letter, not the digit 0). He then recalculated the area with the aid of the binomial theorem, removed all quantities containing the letter o and re-formed an algebraic expression for the area. Significantly, Newton would then “blot out” the quantities containing o because terms “multiplied by it will be nothing in respect to the rest”.

If you read any standard calculus book, you'll see how formulas for areas under such curves are derived, and why they are true.

The equality of the areas of the region you have in mind and the area of that rectangle happen to be a "fun fact", I don't think there's a way to show this geometrically, it's just something that we can observe from the formula we get for the area under the curve x^n.

Also, the area of the rectangle is not the total area under the curve, it's area under the from 0 to x.

Then, if you'd then like to know more, keep watching the videos and there's also some great videos @ www.khanacademy.org to go a little deeper and get a good textbook on the subject. Btw, don't bother with wikipedia it's terrible for beginning to understand a lot (but not all) of math stuff and will scare you away because of the way it jumps from simple to mind bogglingly difficult to simple on about every math topic...