Let's start with the first sentence: "Der braune Hund gehört dem Mann." Translation: The man (a specific man since the definite article (indefinite: einem Mann) is used) owns the brown dog. In the second sentence, we find the preposition "zu". As you might have learned already, zu is a preposition that comes with the dative case (examples for other prepositions: bei, mit, nach, ...). As such, you say "gehört zu dem Mann" rather than "gehört zu den Mann". This sentence expresses that the dog belongs to the man. Therefore, focus is put on the dog (belonging to the man) opposed to the man (owning the dog).