Translation:After one day at the airport, we began to complain.
Here is my GUESS...
I believe začít" is a perfective verb, making it a "one and done" thing. In this sentence, the complaining STARTED once. A different verb (začínat?) would be used to indicate that the complaining started and is continuing, as would be the case with "have begun."
Perhaps a volunteer or more advanced learner will join the discussion to clarify if I'm wrong about this.