‘He answered and said to Arioch, the king's captain, “Why is the decree so pressing from the king?” Then Arioch made the thing known to Daniel. And Daniel went in and desired of the king that he would give him time, and he would show the king the interpretation.'
It was presumably because the captain had had to read out the decree before carrying out the sentence, that Daniel was given time to question him on the matter. So Daniel, who would not have known the full reason for what was happening, asked what pressing matter was causing these summary executions. When informed of the reasons he no doubt asked Arioch to take him to the king, which explains why he was able to obtain access to him. And once there he asked for time so that he could find the answer for him. Nebuchadnezzar clearly accepted the genuineness of his promise for the time was allowed.
‘The king's captain.' Literally ‘the king's captain of slaughterers' (of animals). This may be derogatory suggesting that the captain was acting like a cattle slaughterer, or perhaps the title had attached itself to the captain of the king's guard in some way, just as through history men have been called ‘the Butcher'.