The correctly-punctuated sentence is: "After they rested a few days, the aliens revived sufficiently to explore."
According to English grammar, when you have a sentence with an adverbial dependent clause which comes before the independent clause, you should add a comma before the independent clause.
Here, the adverbial dependent clause is "After they rested a few days:"
- dependent because it does not express a full thought and cannot stand on its own as a sentence,
- adverbial because it starts with the subordinating cunjunction after, thus giving an information about the main verb of the sentence (rested).