Serious question, what is one of the immediate practical applications if this problem (sums of three cubes) is solved? Not a math person so question might sound stupid.

There aren't any, and the solution itself is uninteresting, but it's possible that techniques developed in the process of solving it will find application in solving other problems (In physics, chemistry, genetics, cryptography...)

Consider the most important function of calculus - the integral. In layman's terms, it measures the area under a graph. Okay - that's a little bit useful, if you care about the physics of moving objects (A bus is accelerating at 2 m/s^2 for 5 seconds, how far does it travel..?)

Yet, if you know how to integrate, a mountain of not-immediately obvious physics problems - say, anything that has to do with electromagnetism (Maxwell's equations) immediately become tractable.

I think this falls under the mathematical category of "fun things you can do with numbers", and not much else. Just cool.

Unless in the future someone finds a useful application for this. It's happened before.

Interesting. Like what?

Pretty much all of mathematics was founded on "fun things to do with numbers", and then later we came up with using prime numbers for cryptography and finding Fibonacci sequences in nature, etc.