What does $1^n$ mean?
Is this a bit string of length $n$? If so why don’t they write it as $\{0,1\}^n$?


$\{0,1\}^n$ is not "a string of length $n$", it is the set of all strings of length $n$ (formally, the Cartesian product of $n$ copies of the set $\{0,1\}$). $1^n$ is an element of this set; namely the one where all bits are $1$.