Follow the tangent to where it hits zero, and repeat. Which root you land on depends on where you start — sometimes wildly.
Click anywhere to move the seed. Near a boundary, a hair's move sends it to a completely different root.
Click anywhere to drop a seed and watch it fall to a root.
Newton's method is the most natural idea in numerical mathematics: you can't solve the equation, so replace the function by its tangent line — which you can solve — jump to where that hits zero, and repeat. Each step is a linear approximation, and the derivative is what makes it possible.
It works beautifully. Close to a root the error squares every step: three correct digits become six, then twelve. That's why almost every solver in existence is Newton's method wearing a disguise, and why the same idea generalizes to several variables with the derivative matrix in place of the derivative.
The colouring shows which root each starting point ends up at. Near any one root the picture is calm — everything nearby goes where you'd expect. But the boundaries between the regions are not curves. They are infinitely intricate, and every point on one has all the roots arbitrarily close by.
So the method's guarantee is strictly local. Start close enough and it converges spectacularly fast; start somewhere ambiguous and where you land is, for practical purposes, unpredictable. "Good enough initial guess" is doing enormous work in that sentence.