The CLT gives us a way to understand the universe. It gives rise to certain contradictions. On its own, it is pretty meaningless outside mathematics, but in combination with some beliefs, it can lead to other beliefs.
For example, we can say, "human heights for people in this country are roughly independent (if we consider 30 year olds alive today) and identically distributed (if we choose people at random and we don't have a bias, say, for choosing people in a particular suburb), and that the variance of heights is finite. The 300 people on this plane (and that's probably not great for my assumptions) have an average height of 5 feet and 6 inches. Therefore, the average height for people in this country is therefore near 5 feet and 6 inches or more, or my assumptions are wrong, or this plane is full of unusually tall people. I don't think my assumptions are very wrong, so I think that either the average height is around 5 feet 6 inches or more, or this plane is full of unusually tall people."
Without the CLT, we wouldn't be able to make that inference. It would be perfectly possible that the people on the plane weren't unusually tall, even though the average of their heights is more than the average height for the population. (Obviously, it's hard to imagine a counterfactual to a rule of mathematics, but imagine there were a small number of really, really tall people, none of whom were on the plane.) The CLT removes that possibility, leaving only the possibilities of bad assumptions, fact about the world, or unlikely data.