Othello is most upset that he and his wife have been disrupted on their honeymoon night. He has yet to consummate his marriage with his new bride. Also, his newly-appointed right-hand man, Cassio, fails in his duty regarding a basic assignment: the watch. This obviously makes Othello look bad, especially since Montano, the governor of the island, is injured on his watch.
The monologue is aimed square at Cassio, especially the end. According to "Shakespeare Made Easy," a line-by-line translation is as follows:
"Now by heaven, my self-control begins to be overcome by my anger, and my best judgement is darkened by my passions. By God, if I so much as move or raise my arm, the bravest men among you shall fall before my rebuke. Tell me how this terrible brawl began and who started it, and whoever is convicted of this offense--even if he were my own brother--shall lose my support. What do you think you are doing carrying on a personal and private quarrel in public in a town that is stirred up from war, the people's hearts still full of fear? It's monstrous. Iago, who started this fight?"
In terms of what the words mean, these lines mean something like "I swear that I'm getting really angry. My blood (anger) is overruling my common sense, my passion is starting to lead me (tell me what to do).
The lines you mention are spoken by Othello to Iago, Montano and Cassio. The reason that Othello is so angry is because the three guys, along with Roderigo, have been fighting. Othello can't get answers about why the fight started and he is getting annoyed. So he is starting to tell them to tell him the truth already because he's getting mad.