$\gdef\d{\operatorname{d}}$
There are two ways of thinking about this definition. One is that it is the complex analog of the usual notion of a differentiable function of one real variable. So insofar as the complex numbers, with its algebraic operations and its notion of absolute value, the notion of a holomorphic function is like the notion of a real differentiable function.
But this perspective hides the magic of holomorphic functions, which in many ways are very different from real differentiable functions. To bring out the magic, it’s helpful to think geometrically, viewing the complex numbers as the plane $\R^2$ and thus thinking of $f$ as a function of two real variables.
Then we can ask, what is it that distinguishes a holomorphic function from an arbitrary real differentiable function of two variables? The answer can be phrased as follows. Suppose give a real differentiable function $f: U \to \R^2$ and a point $p \in U$. The derivative of $f$ at $p$ is by definition a certain linear map
$$ \d f|_p : \R^2 \to \R^2 $$
where we think of the source (resp. target) as the vector emanating from $p$ (resp. $f(p)$). Then an immediate comparison of the definitions shows that $f$ is holomorphic if and only if $\d f|_p$ is given by multiplication by some complex number (which will be $f'(p)$).
Now, it is only certain linear maps which are of that form. Multiplication by a complex number is either the zero map, or it is the composition of a rotation and a scaling. A particularly vivid consequence is that when $f$ is holomorphic, $\d f|_p$ is either zero or it preserves oriented angles. Another consequence is a certain rigidity: the whole linear map $\d f|_p$ is determined by its value on any nonzero vector, since already that fixes the amount of rotation and scaling undergone by the whole map.
To bring this to life, recall the geometric interpretation of $\d f|_p$: if you leave the point $p$ travelling with instantaneous velocity $v$, then your image under $f$ will leave $f(p)$ travelling with instantaneous velocity $\d f|_p(v)$. Thus when $f$ if holomorphic, it will preserve (oriented) angles to first order, since $\d f|_p$ preserves angles. In fact this is almost an equivalent condition: a real-differentiable function with non-vanishing derivative is holomorphic if and only if it is conformal, i.e. preserves angles, or even just if and only if it preserves right angles. That’s because the only linear maps which do that are compositions of rotations and scaling, hence given by multiplication by a complex number. Furthermore, there is the rigidity: the first-order movement under $f$ in any direction determines its first-order movement in every direction. e.g. $\d f|_p(i)$ is the $90$-degree counterclockwise rotation of $\d f|_p(1)$. In coordinates, this is known as the Cauchy–Riemann equation.
Moreover, if you imagine fixing the length of $v$ but sweeping out all possible directions evenly, your image under $f$ will also have fixed length and sweep out all possible directions evenly. Thus, a holomorphic map is balanced (the technical term is harmonic).
This first balanced has a large scale manifestation, the mean value property.