Consider a function, $f(x)$, and its plot on a graph ($f(x)=y$). This plot is usually represented by a line, e.g. $f(x)=x$ is represented by a straight line. Now consider a section of this line, e.g. $f(x)=x$ for $0 \leq x \leq 10$. Measuring the length of this line is trivial by using the Pythagorean Theorem or the Distance Formula, etc.
Now consider a graph of a semicircle. In this instance, we can also find the length of the line: we use the radius to find half the circumference.
Now consider a different graph, e.g. $f(x)=x^2$. I can think of no direct way to find the length of this line for some interval on $x$. Is there a way?
I can propose a way, which, to me, seems intuitively related to Riemann Integration. In Riemann Integration, we try to approximate the area under the graph for some interval by fitting arbitrarily small rectangles between the curve and the x-axis. Similarly, one ought to be able to find the length of a line on some interval by fitting arbitrarily many right-angled triangles with equal width on the curve: this way, we can partition a curve into arbitrarily many lines that are approximately straight, which we can measure by finding the hypotenuses of our triangles. Then to find the length of the curve, we just sum those hypotenuses. Diagrams for illustration below:
As stated: in each image, the approximated length of the line is the sum of the lengths of the green straight lines. Clearly, as we let the size of the triangle base $a$ become arbitrarily small, our approximation of the length of the curve will become more accurate.
On a final note, obviously this method has something in common with how we 'intuitively' take a derivative.