Did Luffy's balls break? No, in the scene in the manga it looked like Kaido hit Luffy's vital part, but in fact it did not cause such serious damage. However, careful netizens discovered this scene and took it out, which led to comments such as "Luffy's balls are broken", and there are pictures to prove it, which is really hilarious.
Of course, generally speaking, "the balls are broken" is just a word to describe one's troubles and helplessness. It usually means self-deprecation or self-deprecation. Just kidding, not actual physical harm.