The Residue Theorem in complex analysis is quite impressive and has profound implications for many applications and integral calculations. The ability to reduce the closed integral of a complex function $f(z)$ (or its Laurent series representation $\sum_{n=-m}^{\infty}a_{n}(z-z_{0})^{n}$) in a domain around, say, an $m$th-order pole $(z-z_{0})^{m}$ to just a constant number ($i2\pi$) multiplied by the coefficient ($a_{-1}$) of the first negative power in Laurent series is a massive computational feat.
But what is more baffling to me is: why, of all other coefficients, did $a_{-1}$ turn out to constantly be the only important one? What is the significance of this specific term (namely, $1/(z-z_{0})$) compared to the others or different orders?
I understand how it was derived mathematically in a step by step fashion and how all terms eventually cancel out except for this term. But I just cannot help to wonder about its deeper significance physically or mathematically, when we look at any arbitrary complex function integral in a model or an application and end up just caring about its $a_{-1}$ coefficient (residue) and nothing else? What is the physical or geometrical or mechanical or "real-world" meaning of the residue that can be more intuitively felt in practice?