This sentiment is analyzable by studying three specific fallacies:
Argument from authority
Argument ad populum
The first fallacy is an example of "examining the person instead of their argument" because it makes an appeal to the authority of somebody without actually analyzing the words they are saying. An example would be a chemist telling you that hydrogen bonds are an example of covalent bonding. This is false and believing the chemist purely based off of their status as a chemist would be fallacious. The truth of the statement is not determined by the person's job or education level; the truth of the statement is determined by the facts of the situation. This means you should not evaluate them as a person, instead you should evaluate their words (their argument).
The second fallacy is an support of the proverb because it makes an appeal to popular consensus. For this example, imagine that the year is 300 B.C.E. The example would be: "many people believe that the Sun orbits the Earth, therefore the Sun orbits the Earth." This is a fallacy because it is an appeal to the amount of people that believe a statement instead of the actual facts relating to the statement. The reasoning goes like this: Many people believe this, it would be improbably for a large amount of people to believe something false, therefore it must be true. You can see that this argument has nothing to do with the actual statement, it is merely an appeal to the large amount of people that believe it. It supports the proverb because it shows the fallacious nature of only looking at the amount of people speaking and not at their actual words.
Finally, the third fallacy supports the proverb because it shows the irrelevance of attacking a person instead of their argument. Suppose that Alice and Bob are having a discussion about which flavor of ice cream is the best and Alice says chocolate. If Bob then replies "Well, you voted for Nixon so what do you know?" that would be an example of an ad hominem fallacy. The truth, or merit, of whether or not chocolate is the best ice cream has nothing to do with whom Alice voted for. Bob is trying to show that Alice's opinion cannot be trusted because she made a poor decision in the past but, again, this is irrelevant to the topic at hand. It is not impossible for someone who showed poor judgement once to show good judgement later and whom Alice voted for has nothing to do with the qualitative values of flavors of ice cream.
See also: genetic fallacy