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In invocations like Agonizing Blast, PHB states

Prerequisite: Eldritch Blast cantrip

When you cast Eldritch blast, add your Charisma modifier to the damage it deals on hit.

However on invocations like Armor of Shadows, PHB states

You can cast Mage Armor on yourself at will, without expending a spell slot or material components.

My issue is while Agonizing Blast explicitly requires the spell in the first place to work, Armor of Shadows doesn't state you need to know the spell before gaining the invocation. Will gaining Armor of Shadows give you Mage Armor for free?

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Your interpretation is correct.

Invocations like Armor of Shadows and Eldritch Sight do not require you to already know the spell, and instead let you cast it for free. If you did have to know the spell beforehand, it would be listed as a prerequisite.

Warlocks can't learn those spells otherwise

It's actually impossible for a single-classed warlock to know those spells, given that Mage Armor and Detect Magic are not on the Warlock spell list (PHB 210)

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