Complexity theory is fascinating in itself (talking about how efficiently solvable or complex a problem is in an absolute, mathematical way), but also can spur a lot of research. As an example, integer factoring is strongly believed to be hard for classical computers (in a well defined complexity theory sense). When Peter Shor presented an efficient quantum algorithm for integer factoring in 1994 it was a breakthrough for the field and triggered huge interest into the field of quantum computing and quantum alogrithms. Serious widespread research into quantum computing more or less started then. Quantum computing is estimated (by BCG) to create a market value of 450-850 billion $ by 2050 (mainly in chemistry, optimisation and cryptography).

So, the academically interesting aspect aside, it can spur research for useful things, it can act as a guide to what is worth / not worth researching (if something is proven to be in a "hard" complexity class, then you won't find any efficient algorithm to solve the problem, ever (assuming P =/= NP).) and is generally a great framework for algorithms research. And evidently, algorithms create a lot of value (good and bad, unfortunately).

It's just very much "under the hood". 100 years ago people could have also, rightly, said "how does quantum theory benefit people, what's the value in that". Yet, without the fundamental research in quantum theory we would not have modern technology. Understanding the behaviour of electrons in materials on an atomic level requires quantum mechanics.