In addition, note that in English, a race/culture adjective can be used as a noun for the people or the language. The word "Mexican" in English can be used as an adjective to describe any noun ("Mexican food"), as a noun meaning a person from Mexico ("He is a Mexican"), or as a noun referring to the dialect spoken in Mexico ("Is Mexican different from Spanish"). Similarly for aliens: Klingon can be used as an adjective ("A Klingon ship"), a person ("The captain is a Klingon"), or the language ("Can you speak Klingon?")
In Klingon, the word tlhIngan is a noun and refers specifically to the beings. Nouns can be used in a similar manner to how English uses adjectives and go before the noun they are describing. tlhIngan as a noun does not refer to the language. But tlhIngan as an adjective can be attached to the word for language to specify that it's "the Klingon language". In English we might just use the noun "Klingon", so tlhIngan Hol can be translated as "Klingon language" or as just "Klingon".
In English, singular nouns must have an article. You could also say, "A Klingon language is beautiful," or, "Klingon languages are beautiful," but you cannot say, "Klingon language is beautiful." I suppose an argument could be made that language can be used as a mass noun, but I'm going to stick with treating it as singular and requiring the article.