Learn Russian: У этих опасных собак строгий хозяин. - These dangerous dogs have a strict owner.

This sentence uses the standard Russian way to say "to have": У + [owner in genitive] + [object owned in nominative]. Literally, it translates to "At these dangerous dogs [there is] a strict owner."

Why does этих end in -их but опасных ends in -ых if they are both genitive plural?

The demonstrative pronoun этот (this/these) takes the soft genitive plural ending -их, making it этих. Most regular adjectives with a hard stem (the consonant before the ending), like опасный (dangerous), take the hard genitive plural ending -ых, becoming опасных.

I see the adjective and pronoun have plural endings, but why doesn't the noun собак have a visible genitive plural ending?

Most feminine nouns that end in -а in the dictionary form, like собака (dog), drop that -а to form the genitive plural. This leaves the word with what we call a "zero ending," making it just собак.

Why is строгий хозяин in the nominative case? Shouldn't it be the direct object (accusative case) since the dogs "have" him?