Kakeya problem is a beautiful question, and Besicovitch's construction has been really useful in harmonic analysis (Fefferman's The multiplier problem for the ball is a classic example). Still, in the eyes of wider public it must appear uncanny, requiring the Amazon truck analogies, etc.
Reminds me of Vladimir Arnold's comment on the state of modern number theory (presumably talking about density of primes in arithmetic progressions, he says): why would you even want to add primes, they were born to be multiplied?
If I understand correctly, the initial appeal goes back to the early days of measure theory.
Say, a 2-dimensional set is cut parallel to y-axis, and the cuts all have length 1. If the x-coordinates of the cuts themselves have length 1, you know that the 2-dimensional set must be "large". This is because you can integrate the cut length, and tell that it has area 1. (Think of a 1x1 square for an illustration).
In Kakeya's problem, the cuts still have length 1, but are no longer parallel to any one axis. Besicovitch's construction shows, Kakeya set can have very small area, yet contain cuts of length 1 in many directions. This situation is quite different.
This counterexample turned out to be useful in other areas of pure math, some discussed in the wiki entry for Kakeya problem.
I'd speculate it's similar to the Collatz conjecture, where the problem can be explained simply even to non-mathematicians, but trying to prove it or fully understand it requires a much deeper understanding.
Reminds me of Vladimir Arnold's comment on the state of modern number theory (presumably talking about density of primes in arithmetic progressions, he says): why would you even want to add primes, they were born to be multiplied?