What is exactly meant by this function? I haven't found any clarification how exactly this function works:
$$f: \{0,1\}^n \rightarrow \{0,1\}^m$$
So it takes a n-bit string and turns it into an m-bit string but what is meant by a bit string and what are some examples of this function? Is a bit string 010101010101? This then gets put into the function and this turns into another bit string?
Bit-strings just represent subsets of sets where a $1$ means the corresponding element of the set is included in the subset, and $0$ means it is excluded from your subset.
So if you have the following set: $\{2,3,5,8,13\}$ then the first bit in our bit-string corresponds to 2, the second to 3, the third to 5, the fourth to 8 and the fifth to $13$.
e.g. if you take a bit-string defined as $10110$, this represents a subset of our previous defined set as : $\{2,5,8\}$