It is helpful to read the aphorism from which the quote is coming (Twilight of the Idols, Skirmishes of an Untimely Man, aphorism 26) in full. Let me cite according to the Cambridge Edition (2005):
We stop valuing ourselves enough when we communicate. Our true experiences are completely taciturn. They could not be communicated even if they wanted to be. This is because the right words for them do not exist. The things we have words for are also the things we have already left behind. There is a grain of contempt in all speech. Language, it
seems, was invented only for average, mediocre, communicable things. People vulgarize themselves when they speak a language.
The quote, and the aphorism it is part of, express the idea that there are such things as "true experiences", like immediate moments of being-in-the-world, and that at the moment we try to express them and say something about (us in) the world, we already lessened them by imposing our conceptual structures (words) on them, which distorts them. That is why he speaks of "contempt" since by searching for and using words, we do belittle the truth and elevated importance of true experience, which can only be felt.
About the translation compared to the German original
As an important part of the interpretation, I will provide insights into more literal translations because they provide some contextual meaning and connotations which make it easier to grasp the thrust of the whole aphorism. I understand that it is very hard to translate an author like Nietzsche and not lose a lot of the puns and connotations but in that case, I am afraid they did a particularly bad job. Sentence by sentence:
- My translation would be "We do not value ourselves enough when we talk about ourselves anymore". This has two reasons: Firstly, it introduces the theme "life is prior to language", ie. we can only talk about that which we lived through first, which is taken up in sentence number five. Secondly, it already introduces the main theme, as is paramount to an introductory sentence of an aphorism, which is that there is a valuable true self (or life) that is not valued or misconstrued at the moment we try to stop being (or living) it but talk about it instead. That way, we close the circle in the last sentence (like the Cambridge translation does as well), but also highlight that it is not about communication or language as such, but about the impossibility of meaningful communication about something.
- In the second sentence, the translation basically gives the contrapositive of the original text, which IMHO slightly distorts the meaning. Nietzsche writes in German that our true experiences are by no means loquacious, which more succinctly points out that they do not express or disclose themselves in words. It is not their form of appearance. This is a phenomenological point he is making here, like in the following sentence. And it is categorical, not about the quantity. Another important point is that he does not write about "Erfahrung" but about "Erlebnis", both of which are translated as "experience" in English. In German, they do have different connotations, though: While Erfahrung is exactly that experience that has been made and is present cognitively - can be retold and applied in future experience - Erlebnis is a one-off bodily experience that cannot adequately be captured by words (or even relived), like an epiphany (but without religious connotations). He even writes about the "eigentlichen Erlebnisse", which means those bodily experiences which are to be called "Er-leb-nis" - there is the verb "leben" (living) in there - in the proper meaning of the word, further differentiating it from "Erfahrung".
- In the third sentence, the original does not use the passive, but the active: "They [true experiences] could not talk about themselves even if they wanted to". It is not that we lack adequate words, it is in their nature and even if they could use language, they would not be able to talk about themselves adequately.
- Like in the sentence before, the following one is written more in the active: "That is, because they lack the words." Which basically says the same as "There are no words for them", not only not the right ones. But it also highlights that it is not on us to find words for them (as in "about" them). This would be alien to them since we would impose our words on them.
- The next sentence is a bit tricky. More literally, he writes that we are beyond the things we have words for, ie. we, our life, our being, has moved on and is occupied with other things so that there is no immediate access to the experience anymore. That is also why it is true that we have "already left them behind", but I think the original highlights how we (our true selves) did not "let go" of them. It is not an act of leaving something, it is rather that only when we have moved on and thus are not living the experience anymore, we can conceptualise and start to describe it. It is not exactly like we chose true experience to come and go as we please. Interestingly, the translation in your question is quite metaphorical but very close to the intended meaning (see also the remarks on "Erfahrung" vs. "Erlebnis" above).
- The Cambridge Edition delivers a literal translation here. I, personally, like it better than the one given in the question since there is a meaningful difference between "a grain of contempt" and "a kind of contempt". He does not write about "a kind of contempt", like maybe something similar to contempt, but actual contempt, even if it might only be in small part.
- Nietsche does again use a slightly different formulation: He writes that language was invented for "the average, the mediocre, the talkative". This is, on one hand, obviously a reference to the sentences two and three. True experience is not talkative (mind the active!), hence language was not invented for true experience. But it also provides a bridge to the following sentence since it implies that those people who speak (much) are average and mediocre, which is exactly what vulgar originally and in that time means. Talking about "things" that are "communicable" does not cover all this at all.
- A literal translation is "Already by using language the speakers vulgarise themselves". That highlights that the mere act of using words as a means of self-expression itself is already a vulgarisation of their true, living selves. Thus, even if the translation would have used "whenever" instead of "when", it would have captured the full thrust of the sentence much better.
For context: aphorism 24, art, and life
This inadequacy of words in the communication about the self is also a theme in aphorism number 24:
What is it about himself that the tragic artist communicates? Doesn't he show
his fearlessness in the face of the fearful and questionable? - This in itself
is a highly desirable state; anyone who knows it will pay it the highest
honours. He communicates it [ie. his state of fearlessness], he has to communicate it, provided he is
an artist, a genius of communication. (bolded mine)
Here, the artist does not exactly communicate about themselves, but something about themselves, ie. a highly desirable state they find themselves in. They do not describe or talk about the feeling, they directly communicate the feeling itself (even if through words) and thereby live (embody) it at the same time. Now, what do they, the artists, communicate more generally?
Is the artist's most basic instinct bound up with art, or is it bound up much more
intimately with life, which is the meaning of art? Isn't it bound up with
the desirability of life? - Art is the great stimulus to life: how could art be
understood as purposeless, pointless, l'art pour l'art?
Here, we are given two insights that are relevant to our interpretation: Firstly, that which is paid "highest honours" (the contrary of contempt) is "desirability of life", ie. desirable states in life. Secondly, only the artist - a genius (the contrary of the vulgar) - is able to communicate something of themselves, a desirable state, directly, through the work of art. In other words: only art is able to communicate something meaningful.
Obviously, aphorism 26 is claiming the ineffability of life more categorically, while number 24 allows for some meaningful communication of life through language (not about life!).
I already did some interpretational work, obviously. But it always helps to summarise the ideas first and bring them together with the problem at hand afterwards.
Nietzsche does maintain here that for all the things that are true and important (in life, for ourselves), there can be no words:
This is because the right words for them do not exist.
Even worse, all words are, as we have seen above, for the vulgar only. So where does contempt come into play? I would argue that this contempt has a twofold meaning: On one hand, it is contempt towards true experience (desirable states of life), since it does belittle their exalted status and unique nature. Every try of imposing a language on them necessarily corrupts them. It misconstrues them as ordinary and of lower order - which is what contempt is all about. On the other hand, the same can be said on all humans that use language (for talking about something, not in art). As long as they occupy themselves with language, they neither are open to true experience nor do they act and realise their own potentials. Rather, they actively hold themselves down on a level of vulgarity by talking about the world instead of living in and changing the world itself.
They are, one could say, only in their heads and representing the greatness of past lives. Since life is all about being in and living desirable states, the contempt is contempt for [what makes] life itself [liveable]. This fits nicely with his derogatory remarks on Schopenhauer (letting go of all willing/desire) in aphorism number 24 as well.
That is where the two aphorisms seem to stand at odds: While 24 does allow for the use of language as a means to communicate desirable feelings, 26 does seem to devalue all uses of language. That may be due to the nature of language: it is a mediation that involves an inherent aboutness. But even if an artist uses language, they do not talk about the experience (from the outside), they do not "speak", since speaking is "about" something. The artist lives the experience in producing the work of art, which is why art is able to convey desirable feelings at all. It is the work of art itself, not the things (for which there are words) described therein, that conveys true experience. Seen that way, we could reconcile them.
And since this answer is about what Nietsche wrote (even if I share the intuition behind the words and tried to convey it first and foremost), it is probably an embodiment of vulgarity (sic!) according to him.