the key is to change the denominator so that all of its digits are nines. The approach that your book used was to first change it so that all the digits were one, and then multiply by nine. So we have the denominator 239, and we need to change it (via multiplication) to a number whose digits are all ones.
so we find 1111111 is a multiple of 239, hence:
now, in order to rewrite the fraction with a denominator of nines (so that we can write it as a repeating decimal) we multiply by 9/9:
finally, we convert to the repeating decimal which will follow the usual pattern: