As a (bio)chemist, this paradox has always bugged me profusely.
Perhaps I should set aside some time someday to grasp a very basic understanding of what's going on here. :p
Once I understood this, I find it is a bit a cheat (as you might expect). The 'cuts' are not cuts in any real-world sense.
Hopefully you are happy with the idea that there are as many integers as there are both even integers or odd integers. Therefore in some sense I can "cut" the integers into two equally sized sets. This paradox does something very similar on the real numbers, making use of the form of the real numbers to avoid the "doubling" effect I got when cutting the integers.
You're right that these aren't "pieces" in the intuitive visual sense. That's why Feynman ridiculed the result, because in the initial description given to him he assumed physical pieces, not pairwise disjoint sets.
There are similar paradoxical dissections of the plane:
I remember being told about the library that has all finite books that can possibly be written. Every finite string of characters appears as a book. Obviously there are infinitely many, but we won't worry too much about that.
Now we find that someone, overnight, stole all the books except for those that start with the letter "q". We're annoyed at first, but then someone suggests we just erase the initial "q" from all the remaining books.
So we do that, and lo and behold! We have the full collection again! We even now have an extra book, one that's completely blank!
It's not a paradox, but it is the way infinite things can work, and it shows how 1/26 of a collection can be the same as the whole collection. This is related to how the B-T "paradox" works, but it's very much over-simplified.
Even so, it's a useful place to get people starting to think about things in the right way.