"Cut off" means to interrupt, to stop someone from doing or saying something. "Anna didn't like what Steve was saying, so she cut him off" means that Anna stopped Steve from talking. "He had too much to drink so the bartender cut him off" means that the bartender wouldn't serve him any more お酒.
"Cut in" also means to interrupt, but not to stop. "As Steve was explaining, Anna cut in to ask another question." This means that Anna interrupted in order to ask more information. "When Prince Charming saw Cinderella dancing with another man, he cut in" means that the Prince stepped in between Cinderella and the other man, and began dancing with her.