The verse "Now when the king saw that they pressed him sore, being constrained, the king delivered Daniel to them:" from Bel and the Dragon 1:30 means the following:

The king was under pressure (pressed him sore): The people or priests of Bel were intensely pressuring the king to do something. "Pressed him sore" means they were relentlessly and forcefully demanding action.
He was constrained (being constrained): The king felt forced or compelled to act in a certain way. He didn't necessarily want to, but circumstances (likely fear of the people or political expediency) left him with no other perceived choice.
He delivered Daniel to them: As a result of the pressure and constraint, the king handed Daniel over to the people or priests who were demanding it. This likely meant Daniel was to face punishment or even death at their hands.

In simpler terms: The king was in a tough spot. The people/priests were really pushing him to do something about Daniel. Feeling like he had no other option, the king gave Daniel up to them, probably for punishment.

Context within Bel and the Dragon: This verse appears in the apocryphal story of Bel and the Dragon, which is an addition to the Book of Daniel. In this story, Daniel exposes the trickery of the priests of Bel, who secretly consume the food offered to the idol. This infuriates the people, who demand that the king hand Daniel over to them.