Steve Lee translates 近朱者赤，近墨者黑 as
He who stays near vermillion gets stained red, and he who stays near ink gets stained black.
I matched the Chinese words to the English.
近 = near
朱 = vermilion
者 = he
赤 = red
墨 = ink
黑 = black
But this has left behind "gets stained"! Why doesn't any of the 6 Chinese characters match gets stained? Why did gets stained get lost in translation?
In ancient chinese course, we've learned that it is a phenomenon called 词类活用. So red here means become red, or simply reded. Same for black->blackened.
You'll see so many other examples in ancient chinese(it's much less in modern language but still exists, especially in online buzzwords), where nouns and adjectives turn to verbs and adverbs. I tend to make an analogy with the phenomenon in English where we put some passive voice or perfect tense of a verb as an adjective. (like excited, amazed, and so on) It's like you remove the action from the verb and it become a state. What happens in chinese is exactly the inverse, we add the action(or the history, the motion) to a state(nouns or adjectives) and make it alive. (Attention there is no inflection in chinese, so you have to guess what's the most possible explanation for a particular word with respect to the context. It's hard at the beginning, but after you'll find it necessary especially for ancient chinese reading and writing. Otherwise, the written language will die.)
I'm just an amateur of linguistic so i'm not sure how the experts address my observation, but that's it.
朱 here refers to 朱砂 (cinnabar)
墨 here refers to 墨硯 (inkstone)
者 here means "the one who"
近朱者赤 = the one who gets near cinnabar gets stained/ becomes red
近墨者黑 = the one who gets near ink gets stained/ becomes black
The expression 近朱者赤，近墨者黑 is an analogy for "People get influenced by people around them"
If you are around moral people, you tend to become a moral person yourself; If you are around immoral people, you tend to become an immoral person yourself
In short -- 1. choose your friends wisely. 2. looking at the people around someone and you can tell what kind of person that someone will become 3. get close to bad persons and you will become one
Since it is an analogy that can be applied to different situations, a simple translation of "get stained" is just not practical. It is just one of this expression's implications
This expression is similar to If you lie down with dogs, you get up with fleas in English.
@joehua already answered correctly and directly your question. I would just highlight two things, hoping you are interested in learning Chinese.
近朱者赤，近墨者黑 was written during the Jin Dynasty (265-420 CE).
Even in modern Chinese, there is a structure called "Adjectival Predicate Sentence", that is a sentence with an adjective as the main element of its predicate. This type of sentence describes the state which a person or thing is in. No verb is used to join the predicate with the subject.
Just to make the explanation super-easy for you (I wish), consider the following example:
他好，你不好
Subject: 他, pronoun
Predicate: 好, adjective
Subject: 你, pronoun
Predicate: 不好, adjective
Literally: He good, you not good
As you can see, there is no verb, as in 近朱者赤，近墨者黑:
Subject: 近朱者, noun
Predicate: 赤, adjective
Subject: 近墨者, noun
Predicate: 黑, adjective
In 近朱者赤，近墨者黑, there are no verbs. Verbs are omitted. To translate them to proper English, Steve Lee added "gets stained", which is not found in the original sentences.
Quote:- Why doesn't any of the 6 Chinese characters match gets stained? Why did gets stained get lost in translation?
It is there, but "hidden" in the word "者" Let me explain.
It is feasible, grammatically speaking to substitute "get stained" by transposing it to the "者" when used as a suffix, like the past tense indicator "-ed" as in, say, "suffer(-ed)", rather than the 者 as a "he"
So, we have literally speaking:-
近-Close to, 朱(者)-vermillion(ed),赤-red; 近-close to, 墨(者)-ink(ed), 黑-black.
Therefore Steve Lee probably thought that using "-ed" somehow does not sound "nice" in an English translation, and certainly using "get stained" for the suffix 者 sounds a lot more "poetic" than "vermillioned" & "inked"
Finally, it is the stylistic nature of Chinese idiomatic / poetic expressions to employ economy of words to the extreme, which may be problematic when doing a mere direct translation. So, Steve Lee needed to add "He who..." to make sense of the sentence when those words were not there as well.
百度：
“近朱者赤，近墨者黑”意思是：接近朱砂的地方变红了，接近墨水的地方变黑了；其常用来比喻：和好人在一起，会使人变好；和坏人在一起，会使人变坏。这句话强调的是环境对人的影响很大。
“近朱者赤，近墨者黑”出自傅玄所写的《太子少傅箴》，原句如下：故近朱者赤，近墨者黑；声和则响清，形正则影直。
"声和则响清，形正则影直" 的意思：
声音是悦耳和谐的，那么它听起来就很清越；身形是端正的，那么影子看起来就是直的。
声音是悦耳和谐的，
(if) a voice is sweet-sounding and melodious,
那么它听起来就很清越；
then (one) can hear (it) clearly,
身形是端正的，
(if) a figure (person) is upright
那么影子看起来就是直的。
then (his/her) shadow is straight.
Your confusion is rooted in "false expectation".
The ancient Chinese writing (古文,文言文) emphasis briefness of sentence, and sentences in a sequence must be balanced (對稱), also, often bounded by the requirement of harmonic in rhyming the words (押韻), certain word was dropped, or replaced by another word that does not have the same meaning, of which can be understood only through "knowing the implication of that word in connection of the sentence (會意), and often requires the teaching/interpretation of the person who masters in ancient Chinese writing. So, usually, the ancient Chinese writing/sayings can't be expected to meet today's grammar and be understood easily through analyzing word by word.
By today's standard, the phrase "近(near)朱(vermilion)者(person)赤(red)" does not make good sense, but by "會意", we may guess it means "someone standing near/close to the vermilion likely will be tainted with (沾染) red color". So we may conclude that the verb 沾染 was omitted, and it makes good sense when we understand fully the meaning of the sentence/saying - a person will be affected by the friends he associated with, when close to persons with good traits, the person tends to learn to become a person with good traits too (沾染上好習慣); when close to persons with bad habits, the person will learn and pick up their bad habits (沾染上壞習慣).
