Why should we care about how the world works at an atomic level, if we cannot even see an atom with our own eyes? Why should we care about the origin of our universe, if galaxies and stars both look indistinguishably like shiny dots in the sky to our eyes? And why should we care about developing bizarre theories - such as quantum mechanics - that seem to erase all the intuition we have about the world around us?
At least two different answers to these questions probably exist. The first answer (the naïve one?) would state something like: without knowing how an atom works, we wouldn't be able to predict how molecules form and interact with each other. We thus wouldn't be able to produce most of the chemicals we use in our daily lives, as well as electronic components that keep us glued to the screen of our smartphones. As simple as that: fundamental sciences are needed in order to develop technological progress, which leads to economic development and - someone might argue - to a better way of living. I personally gave the example of physics in the previous answer, but we could repeat this argument across different branches of fundamental knowledge (like mathematics or chemistry) and likely convince ourselves of the same conclusions.
A different way of approaching the initial question leads to a second, longer answer. Humans develop knowledge fundamentally through their senses, and thus can only have access to a limited amount of information out of the total information available in the universe.
How do we understand whether what we observe is not simply a random sequence of events, but events that follow some kind of law? Through abstraction. Starting from an observation, some simple assumptions (like "there exists one and only one line connecting any two points of space"), and a language that allows us to quantify (that is, mathematics), we can build little by little a theoretical framework that explains why we observe what we observe. We can then make our description of reality more complex by extending some laws that seem to work in a specific case to a larger scenario.
A simple example borrowed from mathematics is the concept of raising a number to a semi-integer power. We all know that if x is a positive number and x2 = y, then x = √y. More generally, if xn = y for an integer n, then x = ⁿ√y (try with n=3, x=2, y=8). How can this help us understand what the operation y1/n looks like, for an integer n? Let’s start with another basic rule, that tells us (xn )m = xnm for some integers n,m. It’s therefore easy to convince ourself that if x2 =y, then (x2)1/2 = x = √y and thus y1/2 = √y. More generally, y1/n = ⁿ√y. Crazy, no? By only using a pre-existing framework, we understood the meaning of an operation whose effect in terms of our daily experience seems very blurry, if not impossible to understand. That’s abstraction: starting from a bunch of things we’re already sure of, to build up something new. The previous example applied to mathematics, but this is the actual way all of fundamental sciences achieve greater quantities of information around our world: by asking questions that might at first seem useless, but that ultimately lead to some greater degree of knowledge. Why do we need it, apart from “knowledge for the sake of knowledge”?
Abstraction provides us with the capability of overcoming some limits that are intrinsic to the very fact of being finite human beings. It allows us to somehow achieve a degree of knowledge that would otherwise be constrained and limited by our senses. This is a powerful tool not only when doing science or engineering, but also when confronting ourselves with realities that are different from what we're used to. And we can develop this tool through education. In a world that seems very much fragmented, full of social, economic, and environmental challenges, having the capability of transcending the emotional dimension of these issues is surely a skill every leader acting for the good should have.
Keep from these lines what you wish, and do feel free to disagree. Most of all take the urge to understand how ridiculously bounded we are as human beings, but that not as such shall we act.