Learn Dutch: Ik weet dat mijn mening geen feit is. - I know that my opinion is not a fact.

The word dat (that) is a subordinating conjunction, which is the current lesson's focus. It introduces a dependent clause and forces the conjugated verb to the end. As a standalone main clause, you would say Mijn mening is geen feit, but because it follows dat, the verb is gets pushed to the very end: dat mijn mening geen feit is.

Can I drop the word dat like we often drop "that" in English?

No, you cannot. In English, you can casually say "I know my opinion is not a fact," leaving out the word "that." In Dutch, you must always include dat to connect these two parts of the sentence.

Why does this sentence use geen instead of niet?

In Dutch, you use geen to negate a noun that is indefinite (meaning it uses "a", "an", or no article). Because the English translation is "not a fact", Dutch combines the "not" and the "a" into geen, resulting in geen feit. You generally only use niet when negating verbs, adjectives, or specific things (like "the fact").