Learn German: Ich stelle mir die Zukunft vor.

Because there is already a direct object in the sentence (die Zukunft). When a reflexive verb has a direct object (the thing you are acting upon or thinking about), the reflexive pronoun usually gets bumped into the dative case. So, you use mir instead of the accusative mich.

I thought vorstellen meant "to introduce". Why does it mean "imagine" here?

It all depends on the case of the reflexive pronoun! Sich vorstellen with an accusative pronoun (like Ich stelle mich vor) means "to introduce oneself". But when you use a dative pronoun and a direct object (like Ich stelle mir die Zukunft vor), the meaning changes to "to imagine something".

Why is vor sitting all alone at the end of the sentence?

The infinitive of the verb is sich vorstellen. Because it is a separable verb, the prefix vor breaks off and goes to the very end of the sentence, while the conjugated part (stelle) takes the standard second position.

What case is die Zukunft in this sentence?

It is in the accusative case because it is the direct object—the thing being imagined. Since Zukunft is a feminine noun, its article remains die in the accusative case, so it looks exactly the same as it does in the nominative.