"We cannot wait. We have to leave now."
Translation:Nous ne pouvons pas attendre. Nous devons partir maintenant.
I constructed this sentence as, 'Nous ne pouvons pas attendre. Il me faut que nous partons maintenant,' and the answer said 'Nous ne pouvons pas attendre. Il faut que nous partions maintenant.' Why 'il faut' rather than 'il me faut' and also what tense did Duo conjugate 'partir' in?
Partions is the subjunctive form of partir: don't worry if you haven't learned what subjunctive is yet, there is a lesson dedicated to it towards the end of the tree. Basically it's a special verbal mood required after some expressions including "il faut que".
Note that in this sentence, you don't have to use subjunctive with 'il faut': for instance, an equivalent (but a bit old-fashioned) wording would be "Il nous faut partir" where nous is an object pronoun just like in the construction "il me faut" which means "it is necessary for me [to do smth]".
And since 'nous' has already been mentioned in the previous sentence, you don't need to add it as it's already clear through context that 'nous' is the subject of 'partir', and so you can say "Nous ne pouvons pas attendre, il faut partir maintenant" which is the most natural-sounding solution to me.