Learn Russian: Этот факт не имеет никакого значения.

This is the genitive of negation in action! The verb иметь (to have) is transitive and usually takes an accusative direct object. However, when it is negated by не, the direct object often switches to the genitive case. This rule is especially strict for abstract concepts like значение (significance/meaning).

Why are there two negative words here: не and никакого?

Russian requires "double negatives" (often called negative concord). Whenever you use a negative word like никакой (no/any) or ничто (nothing) in a sentence, you are grammatically required to also negate the verb with не. So, a word-for-word translation is "doesn't have no significance".

Why does никакой take the form никакого here?

It has to agree with the noun it modifies: значение. Since значение is a neuter noun, and we need it in the genitive case due to the negated verb, we must use the neuter singular genitive form of the pronoun, which is никакого. Also, remember that the -ого ending is always pronounced with a "v" sound instead of a "g".

Could I just say "Этот факт ничего не значит" to mean the same thing?

Yes, ничего не значит (means nothing) is grammatically correct and perfectly understood. However, иметь значение is a very common fixed phrase that translates closer to "to matter" or "to be relevant". Saying не имеет никакого значения is a natural, slightly more emphatic way to say "does not matter at all."