The answer to this question is a lot easier to understand graphically, but basically you have to imagine that in , the f(x) is the height of a rectangle and the dx is some arbitrarily small width of that rectangle. The integral means, in essence, adding up all the rectangles - which amounts to finding the total area under the curve.
Edit: this is not really super helpful in terms of answering your question, but is roughly relevant and amused me. Why don't I think of things like this?
Strictly speaking, it's because the integral is what we define the "area under a curve" to be, and the fact it's consistent with the definition of area for simple shapes is what allows this definition to make sense.
the area (called the Riemann integral) is the limit of the riemann sum over the certain interval - end points (a,b), divided into intervals of equal length b-a/l, with a point c in each interval. Sum is then SIGMA i=0 to n-1 of f(c_i)(x_i+1-x_i)