We first need to understand what do we mean by Hamming Distance, simply put, for two strings of equal length, it is the number of different symbols in corresponding locations. For example: since we're exclusively dealing with binary strings, let's say we are calculating Hamming distance between
000. Since only the right-most values differ, we get a hamming distance of $1$.
010 has a hamming distance of $3$ since they are different at each position. Hamming distances are widely used in Coding Theory for Error detection and can also be used in Machine Learning as an alternative measure to Euclidean distance. In both the applications it is used to objectively calculate how separate two values are, more the hamming distance more dissimilar they are considered to be.
Here, hamming distance has the same purpose, to calculate how dissimilar the states are. As you move farther away from $|000\rangle$ (going towards $|111\rangle$) the bigger will be the difference between the states and hence the bigger will be the Hamming distance. Going with the same logic, $|000\rangle$ and $|111\rangle$ have a hamming distance of 3 hence signifying that they are very different (or better said, completely opposite since
hamming distance =