The kick itself did not originate in Brazil, since it's basically a modified roundhouse kick; it saw some use in Kyokushin karate and some schools of Taekwondo but the reason why the kick claims its Brazilian name is because a K-1 kickboxer named, Glaube Feitosa, who has roots in kyokushin karate.
The kinetics of the kick are what makes the kick unique. The Brazilian kick is inherently deceptive; the windup involves faking a low/mid kick then quickly torquing the hips and flicking the shin up to initiate the actual strike to come down on the head, neck, or collarbone. Because of the unusual angle of the kick's trajectory and the initial feint, the Brazilian kick often catches opponents by surprise.
Due to the complexity of the kick, however, it requires a fair amount of set-up work before it can reach its full potential. One of the most common methods for setting up the Brazilian kick is to first condition the opponent to react to low roundhouse kicks or mid front kicks. Once an opponent grows accustomed to blocking the feint kicks to the body or legs, then the Brazilian kick can be used to sneak past the guard and go upstairs to the head or neck, where they should not be expecting it to be.
On a fun note, Jin Kazama in Tekken 4 and onward utilizes the Brazilian kick in his moveset after switching to a more traditional karate style.
