Learn Russian: Я хочу позвать друга.

Because the friend is the direct object receiving the action of being invited. In Russian, masculine nouns for living things (animate objects) take the Accusative case by adding -а to the end, making them look exactly like the Genitive case.

Since this lesson is about Dative pronouns, if I wanted to say 'I want to invite him', would I use the Dative pronoun ему?

No, you would use the Accusative pronoun его. The verb позвать (to invite/call over) takes a direct object (Accusative). This is a great reminder that just because an action involves a person, it does not automatically mean you use the Dative case!

The verb позвать translates to 'to invite', but doesn't it also mean 'to call'?

Yes! It specifically means 'to call someone over' with your voice, which is why it is used for casual invitations among friends. If you want to say 'to call on the phone', you must use the verb звонить. And here is the trick: unlike позвать, the verb звонить actually DOES require the Dative case (for example, 'to call him' is звонить ему).

Why does позвать stay in its dictionary form here instead of matching the 'I' subject?

When you put two verbs together to express that you 'want to do' something, only the first verb (хочу) changes to match the subject. The second verb (позвать) stays in its infinitive (dictionary) form, which works just like English.