It boggles my mind that a number (an uncomputable number, granted) like BB(748) can be "independent of ZFC". It feels like a category error or something.
The number itself is not independent of ZFC. (Every integer can be expressed in ZFC.) What's independent of ZFC is the process of computing BB(748).
I think the more correct statement is that there are different models of ZFC in which BB(748) are different numbers. People find that weird because they don't think about non-standard models, as arguably they shouldn't.