It’s a problem that is easy to state, yet hard to fully solve, and doesn’t seem like it should be unsurmountable, as it is easy to make some progress on (the “integers that equal 4 or 5 modulo 9 can’t be written as the sum of 3 powers of integers” part handles 22% of cases, and interested kids can understand it)

That makes it interesting to most mathematicians (if it weren’t easy to state, it could still be interesting, but to a smaller audience)

Edit: it also is easy for non-mathematicians with a knack for computing to get a crack at. You need not know much about number theory to write a program that searches for solutions for the unknown cases (it will help, but knowing how to optimize programs may help more, and that’s not something all mathematicians are good in). That grows the set of people who might find this interesting.