${var#pattern} is a parameter expansion that expands to the value of $var with the shortest possible match for pattern removed from the front of the string.
Thus, ${VAR#*:} removes everything up and including to the first :; ${VAR#*:*:} removes everything up to and including the second :.
The trailing *s on the end of the expansions given in the question don't have any use, and should be avoided: There's no reason whatsoever to use ${var#*:*:*} instead of ${var#*:*:} -- since these match the smallest amount of text possible, and * is allowed to expand to 0 characters, the final * matches and removes nothing.
If what you really want is an array, you might consider using a real array instead.
# read contents of string VAR into an array of states
IFS=: read -r -a states <<<"$VAR"
echo "${states[0]}"  # will echo NJ
echo "${states[1]}"  # will echo NY
echo "${#states[@]}" # count states; will emit 3
...which also gives you the ability to write:
printf ' - %s\n' "${states[@]}"  # put *all* state names into an argument list