The Cauchy Riemann equations in effect say that a function $f(z) = u(z)+iv(z)$ can be approximated as roughly a scaled rotation
$f(c+h) \approx f(c) + f'(c)h = f(c) + \begin{bmatrix}u_x & -v_x\\v_x & u_x\end{bmatrix} \begin{bmatrix}h_x \\ h_y \end{bmatrix}$
Informally, if $f$ has this kind of approximation at every point in a domain then it admits a power series representation at each point.
Intuitively why should having the scaled rotation approximation give a full power series representation? Suppose at some point $c$ the Cauchy Riemann equations are satisfied but there does not exist a power series representation at $c$, then presumably there should be a geometric point of view to see that this is ridiculous and leads to $f$ not satisfying the Cauchy Riemann equations elsewhere.