What is the motivation/importance of the concept 'holomorphic'?

Briefly, holomorphic functions are immensely useful because, on the one hand, they are surprisingly common (since any power series, for example, whose coefficients grow reasonably slowly defines a holomorphic function), and on the other hand one can prove very strong theorems about them. There is a web of results including Cauchy's integral formula and the identity theorem which assert that holomorphic functions are astonishingly rigid: given information about a holomorphic function in a very small part of its domain, one can extract information about the function's behavior in other a priori unrelated parts of their domain (and this is what allows things like contour integration to work).

For that reason, holomorphic functions are a powerful tool to apply to a problem when they do apply. For example, analytic number theorists frequently construct holomorphic or meromorphic functions which carry number-theoretic information, such as the Riemann zeta function, to prove theorems like the prime number theorem. Since you say you have a background in combinatorics, you might enjoy reading Flajolet and Sedgewick's Analytic Combinatorics, a thorough exposition of (among other things) ways to use complex analysis to provide asymptotics for combinatorial sequences.

Here is a simple example. Let $f_n$ denote the number of ways that $n$ horses can win a race, with ties. It turns out that this sequence has generating function

$$F(z) = \sum_{n \ge 0} \frac{f_n}{n!} z^n = \frac{1}{2 - e^z}.$$

This function is meromorphic with poles at $z = \log 2 + 2 \pi i k, k \in \mathbb{Z}$, each of which has residue $-\frac{1}{2}$. In fact, it turns out that $F(z)$ admits an infinite partial fraction decomposition

$$F(z) = \sum_{k \in \mathbb{Z}} \frac{1}{2(\log 2 + 2 \pi i k - z)}.$$

And by expanding the terms on the RHS in a geometric series, this gives an asymptotic expansion for $\frac{f_n}{n!}$ with leading term $\frac{1}{2 (\log 2)^{n+1}}$. In other words,

$$f_n \sim \frac{n!}{2 (\log 2)^{n+1}}.$$

The pole at $z = \log 2$ dominates the asymptotic expansion: the leading term in the error of the above expression is given by the other poles nearest the origin, which occur at $z = \log 2 \pm 2 \pi i$. Because these poles have nonzero imaginary part, if you plot the error in the above approximation you'll find that it oscillates. It is not so easy to explain why this should be the case without complex analysis.

A famous example is Hardy and Ramanujan's asymptotic formula for the partition function

$$p(n) \sim \frac{1}{4n \sqrt{3}} e^{ \pi \sqrt{ \frac{2n}{3} } }$$

which is proven using a much more sophisticated version of the above argument.

But really, there is too much to say about holomorphic functions, so again I suggest that you read a textbook. Besides Needham's book, I also personally enjoyed Stein and Shakarchi, which is very user-friendly and has good applications.