Dictionary ordering works when you have a set of things and each thing is a string of symbols that have a well-ordering on them.
For words: a word is a string of letters (there are 26) and there is an ordering on the letters: A < B < C< ...
Since the letters are ordered and the words are strings of letters, we can order the words by comparing letters. Consider "baseball" and "bathtub". To order these, we check the first letters: b = b, so no help. We check the second letters: a = a, so no help. We check the third letters: s < t, so we have "baseball" < "bathtub", and so "baseball" appears in the dictionary before "bathtub".
For points in 4-space: points are strings of digits (there are 10) and there is an ordering on the digits: 0 < 1 < 2 < ... < 9.
Since the digits are ordered and points are strings of digits, we can order the points by comparing digits. Consider "(1,2,3,4)" and "(1,2,7,1)". To order these, we check the first digits: 1 = 1, so no help. We check the second digits: 2 = 2, so no help. We check the third digits: 3 < 7, so we have "(1,2,3,4)" < "(1,2,7,1)" in the dictionary ordering.