There are two very common ways this phrase is abused, so that it can be tempting to condemn its use categorically:

But it's one thing to say that one should avoid cumbersome or misleading uses of an expression, and quite another to say that one should never use it, going to lengths to avoid it even when it seems natural to the writer. The latter is arrant pedantry. Sometimes avoiding "the fact that" would be awkward or change the meaning of the sentence:

The fact that Mister Rogers is dead still upsets me.

You could chop off "the fact", and it would still be correct, but it would be a jarringly archaic usage. "Mister Rogers' death still upsets me" says something related, but different: the new version is about an event in the past rather than the state that follows it. The fact that Mister Rogers is dead is not under debate, so there's no question being begged here. And hey, wasn't that just another appropriate use? There are many ways to rephrase that sentence, but they all either sound awkward or change the meaning or voice of the utterance.

In fact, here's a general rule: it's appropriate to talk about the fact of something when it's the fact itself (i.e., an event or state considered as a particular piece of human knowledge) under discussion, as opposed to the underlying reality that the fact is concerned with.