You can use ToRadicals to convert it to the traditional notation with radicals:
ToRadicals@Root[1 - #1^2 + #1^3 &, 1]
Yes. You can write the root of any polynomial of 5 or lower with radicals. But very often you don't want to.
- Sometimes it's just hard to read.
- Floating Point issues. As you may know, the standard quadratic formula taught in highschool can be a floating point disaster. So it's not clear you always want to immediately convert it into an expression made up of radicals.