This is the second half of the Dwarvish battle cry Baruk Khazâd! Khazâd ai-mênu! It translates as "The Dwarves are upon you!", and is the only example we have of a complete sentence in Khuzdul. It doesn't have a verb, but since it has a subject (Khazâd "The Dwarves") in apposition to a prepositional phrase (ai-mênu "upon you) it matches the syntax requirements of a typical non-verbal or "equational" sentence in Semitic languages. It shows that the (probably standard) word order for verbless sentences in Khuzdul is subject-predicate, which is the same as in Hebrew and Arabic.
We are told in quite a few places that Khazâd is "Dwarves" or "The Dwarves". It is the Dwarves' own name for their race, so it could be considered a proper noun. As such, Arabic would consider it semantically and syntactically definite. Translating with the word "the" may be simply a requirement of English, not Khuzdul. This is why we don't see anything we could interpret as a definite article in the sentence. If the speaker is talking about a specific sub-group of Dwarves, I could see the possibility of adding the definite article to distinguish from talking about the entire race in general.
Khazâd is "Dwarves" or "the Dwarves" and is plural, nominative, definite (proper noun) or sometimes indefinite depending on context.
Ai is the preposition "upon", and is a reduced form of aya. It's common in Semitic languages for prepositions to take reduced forms and attach to their objects when the object is a pronoun, which we have here. The final /a/ apparently gets removed, which is similar what we see when -ûn is added to narag to produce Nargûn. This occurs in Adunaic as well. The /y/ then changes to an /i/, which is fairly typical when a /y/ appears in a coda position by itself or immediately before another consonant as part of a coda cluster.
Mênu is "accusative pl. 'you' " according to Tolkien. Comparing with Felak-gundu, Uzbad Khazad-dûmu, and Adunaic's objective genitive, it looks like the -u is the accusative marker. For Khazad-dûm, the final /u/ doesn't get reduced when the accusative -u affix is added, so mên may be the nominative form of mênu. The object of a preposition apparently is placed in the accusative case. This is actually a little different from Hebrew and Arabic. Arabic prepositions make their object take the genitive case, which Khuzdul doesn't seem to have. Hebrew doesn't have a genitive or accusative case per se, so its prepositional objects are in the same form as subjects.
Khazâd ai-mênu! is thus "The Dwarves are upon you!" and is a verbless sentence with a prepositional predicate.
The Lord of the Rings, The Two Towers, Book III, ch 7
The Lord of the Rings, The Return of the King, Appendix F
The Silmarillion, index, pg 337
The War of the Ring, pg 20
The War of the Jewels, pg 209, 387
Parma Eldalamberon XVII: Words, Phrases and Passages, pg 85