"After she cooked the meal, we ate."
Translation:Après qu'elle a cuisiné le repas, nous avons mangé.
Am I right in thinking both après and après que mean after. If so, why does it have to be après que here? Thanks.
It's true that "après" and "après que" both means "after" when it's a temporal indication.
When you use "après" you need something with it, a temporal or physical indication : "après la nuit" -> after the night "après cette maison" -> "behind this house" (even if "derrière cette maison" is better)
In our case it's a temporal indication "after she cooked the meal" so it must be "Après elle a cuisiné le repas". But this doesn't mean anything in french, you can't add a sentence in an other one without a word to link them. This word is often "que" which is contracted in "qu'" here because the first letter of "elle" is a vowel. That's why you say "Après qu'elle a cuisiné,. ..."
You could have said "Après avoir cuisiné, ..." which is totally correct in french and which doesn't need the "que" because "avoir cuisiné" is an infinitive sentence and not a conjugate sentence. But the idea is not totally the same because in this case you don't know who cooked. "Après avoir cuisiné" is more the translation of "after the meal was cooked".
So here, the sentence with "après que" is the best choice.