Because he had changed the will of the king of the Assyrians concerning them …: As the author has already made clear, the new freedom of the Jews in their own homeland was due to the favor of the Persian emperors. The Assyrian Empire had fallen long before this, when the Babylonians conquered them in 612 B.C. However, here the author seems to be thinking of the initial attacks on Israel by Assyria as beginning a time of troubles which only now was being ended. The Assyrian kings had wished them ill, as had the Babylonian kings, but now that has been reversed. This interpretation of the first half of this verse is more likely than that the writer intends to use the king of the Assyrians to designate the Persian emperor simply because he ruled the area once held by the Assyrians (so Good News Bible footnote).

To strengthen their hands for the service of the Lord God of Israel: The Lord is the actor here, not the Persian emperor. This idiomatic clause means to make the Jews serve the Lord in a stronger, more effective way. Now that they were in their own land, the Promised Land, and had their Temple again, they could follow the rituals laid down in the Law of Moses. They were in a better position to serve the Lord.

Here is an alternative model for verses 14-15.

• 14 They also observed [or, celebrated] the seven-day Festival of Unleavened Bread, happy in their worship of the Lord [or, as they worshiped the Lord], 15 because he had ended the hard times they had suffered ever since the Assyrian king had invaded Israel. They were now able to serve the Lord, the God of Israel, more effectively.

Quoted with permission from Bullard, Roger A. and Hatton, Howard A. A Handbook on 1-2 Esdras. (UBS Helps for Translators). Miami: UBS, 2019. For this and other handbooks for translators see here.