"He thinks you know his brother."
Translation:Он думает, ты знаешь его брата.
Animate nouns are things that are alive and can move by themselves, e.g. people, animals. Everything else is an inanimate noun, e.g. concepts, plants, cars.
It's a general rule of grammar, according to https://en.wikibooks.org/wiki/Russian/Grammar/Accusative_case1
Masculine nouns in the accusative case take nominative endings if they are inanimate and genitive endings if they are animate. In simple terms, for Masculine Accusative nouns, inanimate objects do not change their default ending, while such animated nouns either add -а or replace -й and -ь with -я.
Evidently 'что', introducing a subordinate clause, can be omitted following 'думать'. Can this be done also with other verbs of "thinking"? How about 'считать', 'верить', 'воображать', 'предполагать', 'допускать', 'сомневаться', etc.? Can it be done with any non-"thinking" verbs?