What makes so many people refuse to accept Nietzsche's phrase, "there are no facts, only opinions"?

Perhaps because, if taken literally, it doesn't make sense. If there were no facts only opinions, then it would be a fact that there are only opinions and a statement to that effect would be true. But then there would be a fact, so the statement would be false. A statement that is both true and false is nonsensical, and people have every right to refuse to accept a nonsensical statement.

It should be noted, however, that Nietzsche didn’t actually say this; he said there are only interpretations. Moreover, he didn’t intend it to be taken literally. In context, it’s presented as a one-sided riposte to the equally one-sided claim that there are only facts. The truth, as Nietzsche knew very well, is that there are facts and interpretations, that facts must be interpreted, and that the interpretation of a fact is often more important than the fact itself.