The similarities usually cited between Insular Celtic and the Semitic languages and those of North Africa are the following:
- VSO as basic word order.
- "Conjugated" prepositions, where prepositions governing a pronominal object fuse with a clitic, rather than simply being followed by the independent form of the pronoun. E.g. Hebrew לי li "to me" instead of *ל־אני *l-ʾani.
- "Construct" state, where genitive phrases come in the form "[regens] (article) [rectum]" (the regens is the possessed object, and the rectum is the possessor), where the presence of the definite article instead of giving the definiteness of the rectum gives the definiteness of the entire phrase. E.g. Hebrew בֵּית סֵפֶר bēt séfer "a school" (lit. house-of book" & בֵּית הַסֵּפֶר bēt ha-séfer "the school" (lit. "house of the-book").
- A construct state is often described as being a case-like marking of the possessed object in a genitive phrase. By this definition the Insular Celtic languages lack such a similarity.
Unexplained is a massive overstatement. None of these are especially unusual cross-linguistically, and transition to these from more typical Indo-European norms are also well-attested globally.
Vennemann argues for a Semitic (or sometimes some other Afroasiatic) substrate in Celtic and Germanic. His work ignores the fact that these similarities are absent in the earliest stages of the Insular Celtic languages (e.g. Primitive Irish in Ogham inscriptions, and Brythonic inscriptions of the Roman era), and so any contact would need to have occurred during the historical era, and if it was sufficiently intense to cause such major structural changes and in such an era, we absolutely would see evidence of it that we do not. His ideas in this regard are pretty fringe.
It's also worth noting that conjugated prepositions do occur to a limited extent in other Indo-European languages e.g. Spanish con "with" has a full set of conjugated forms like conmigo "with-me".