The idea behind Newton's method is first-order approximation. If $f(x)$ is differentiable at $x_0$, then near $x_0$ it is similar to the linear function
$$ f(x) \sim f(x_0) + (x - x_0) f'(x_0) $$
So if the equation
$$ f(x_0) + (x - x_0) f'(x_0) = 0$$
has a solution $x = x_1$, then $f(x_1) \sim 0$. When everything is well-behaved, this will be a better approximation of 0, and repeating the process will converge to a root of $f(x)=0$.
The same idea works in higher dimensions. If we have two functions $f$ and $g$, then
$$f(x,y) \sim f(x_0, y_0) + f_1(x_0, y_0) (x - x_0) + f_2(x_0, y_0) (y - y_0)$$
$$g(x,y) \sim g(x_0, y_0) + g_1(x_0, y_0) (x - x_0) + g_2(x_0, y_0) (y - y_0)$$
If we set the right hand sides to zero and find a solution $(x,y) = (x_1,y_1)$, then $f(x_1,y_1) \sim 0$ and $g(x_1,y_1) \sim 0$.
(Here, $f_1$ means "the derivative of $f$ with respect to its first variable", etc)