Learn Danish: Han stræber efter at opnå målet. - He strives to achieve the goal.

Because the person striving and the person achieving the goal are the exact same (Han). In Danish, when the subject of the main verb and the secondary action are identical, you should use an at-infinitive. Using a full clause like at han opnår sounds unnatural when the subject is just repeating itself.

What happens if the subjects are different, for example if we want to say 'He strives for us to achieve the goal'?

This is where you are forced to use a full at-clause instead of an infinitive. Because the subjects 'he' and 'we' are different, the subject co-reference constraint no longer applies, and you would say: Han stræber efter, at vi opnår målet.

Is it normal in Danish to have a preposition like efter directly followed by an at-infinitive?

Yes, it is perfectly normal and very common! While English often requires an '-ing' verb after a preposition (like 'strives toward achieving'), Danish prepositions can sit right in front of an at-infinitive. The fixed phrase is at stræbe efter, so efter just connects directly to the infinitive action.

Could I use at nå instead of at opnå in this sentence?