Try to mentally (or on paper) graph the freelancer's pay. The x-axis represents "hours" and the y-axis represents "pay per hour". The function we get is the linear function "f(x) = x". I.e. for the first hour, he's paid $1; for the 300th hour, he's paid $300. A very straightforward graph. If we cut off our x-axis at $300, the graph looks like a right triangle.
If you want to find the total amount of money he's accumulated, all you have to do is find the area under the triangle (which is a special case of integration). For a triangle, geometry class taught us "Area = (1/2)(base)(height)". Calculus courses prefer that we integrate, which has broader application. Integration of "f(x) = x" gives us "f'(x) = (1/2)(x^2)". Our "integral" is equivalent to the "geometry-class equation" because for our particular triangle, the base is the same as the height. For comparison:
Area = (1/2)(base)(height)
f'(x) = (1/2)(x^2)
DoubleMalt is saying kjay's visualized "shortcut" (intuitively faster approximation) is slightly off. This is because the triangle in our actual problem looks like a staircase, and not a smooth slope. But kjay's shortcut has a margin of error of only about "-0.033%" which is a good enough approximation by most people's standards.
Also, DoubleMalt's equation should look like
x * (x + 1) / 2
instead of
x * (x - 1) / 2