Feynman tells the story in one of his books of anecdotes.

http://www.ee.ryerson.ca/~elf/abacus/feynman.html

$12$ is a very good first approximation and the linear term of the series expansion suffices to get high precision.

$$ \sqrt[3]{1728 + d} = 12\sqrt[3]{1+x} = 12 + 4x + O(x^2)$$

where $d = 1.03$ and $x = \frac{d}{1728}$ is, in Feynman's words, about 1 part in 2000, so that the error term is of order $10^{-6}$.

Feynman says that he computed $12 + \frac{4d}{1728}$ as the approximate value.

The number was 1729.03. I happened to know that a cubic foot contains 1728 cubic inches, so the answer is a tiny bit more than 12. The excess, 1.03 is only one part in nearly 2000, and I had learned in calculus that for small fractions, the cube root's excess is one-third of the number's excess. So all I had to do is find the fraction 1/1728, and multiply by 4 (divide by 3 and multiply by 12). So I was able to pull out a whole lot of digits that way.

He describes that as though $d=1$ for this part of the calculation, so maybe $12 + \frac{1}{432}$ was what he actually computed. By "adding two more digits" (to 12.002) he seems to mean working out the division in the fraction. It could also mean adding (0.03)/432 as two more decimal digits of accuracy to $(12 + 432^{-1})$, which requires only a multiplication by 3 of an already computed quantity 1/432.

Feynman's method is the one that would have been immediate for anyone familiar with the binomial series and with $12^3 = 1728$. He said that knew the latter as ft^3/in^3 and other people might know it from the Ramanujan 1729 story. The other ingredient, as Feynman says in the story, was being good at integer division.