Why did Elisha kill the children?

First, the Hebrew word translated "children” (or young lads in the NASB) refers to men who were in their youth, so typically this meant men between the ages of 18 and 30. Most modern translations use the word "youth" in this passage, not children, to reflect that this mob was not a mere group of children teasing an old man. These were young men who knew exactly who they were mocking and what they were doing.

So, we must consider the context of these events to understand why they mocked Elisha and why Elisha responded with a curse.

First, in 2 Kings 2 Elisha had just assumed the role of prophet of Israel in place of the previous prophet, Elijah. Moments earlier, God had removed Elijah from the earth in a spectacular, supernatural manner:

2Kings 2:11 As they were going along and talking, behold, there appeared  a chariot of fire and horses of fire which separated the two of them. And Elijah went up by a  whirlwind to heaven.
2Kings 2:12 Elisha saw it and cried out, “ My father, my father, the  chariots of Israel and its horsemen!”And he saw  Elijah no more. Then  he took hold of his own clothes and tore them in two pieces.
2Kings 2:13 He also took up the mantle of Elijah that fell from him and returned and stood by the bank of the Jordan.
 

A fiery chariot came down and picked up Elijah and took him off to heaven. The chariot was visible to more than just Elisha, because immediately after the event, others nearby offered to go looking for Elijah believing that the charoit would drop him off on earth somewhere nearby.

2Kings 2:15  Now when  the sons of the prophets who were at Jericho opposite him saw him, they said, “The spirit of Elijah rests on Elisha.”And they came to meet him and bowed themselves to the ground before him.
2Kings 2:16 They said to him, “Behold now, there are with your servants fifty strong men, please let them go and search for your master; perhaps  the Spirit of the LORD has taken him up and cast him on some mountain or into some valley.”And he said, “You shall not send.”
2Kings 2:17 But when  they urged him until he was ashamed, he said, “Send.”They sent therefore fifty men; and they searched three days but did not find him.
2Kings 2:18 They returned to him while he was staying at Jericho; and he said to them, “Did I not say to you, ‘Do not go’?”
 

Meanwhile, Elisha knows better than to go looking for Elijah, since he knew Elijah will not be found, so he refuses to join the search party. Later in Jericho the searchers report back to Elisha that their mission failed, to which Elisha says, "I told you so." At this point Elisha leaves for Bethel, and as he is leaving, a gang of young men from Jericho begin mocking Elijah on his way:

2Kings 2:23 Then he went up from there to Bethel; and as he was going up by the way, young lads came out from the city and  mocked him and said to him, “Go up, you baldhead; go up, you baldhead!”
2Kings 2:24 When he looked behind him and saw them, he  cursed them in the name of the LORD. Then two female bears came out of the woods and tore up forty-two lads of  their number.
 

These men are in the same area in which all these events take place, so it is clear they understood what had just happened to Elijah. Notice their chant to Elisha: “Go up, bald head, go up…”  That phrase means they wished Elisha would disappear up in the clouds just as Elijah had gone up earlier. Their chant reveals their hearts. They do not wish to bear the chastisement of a prophet of God. Their sin convicts them, and they want to cast off the restraints of God’s word delivered through the prophets. They were pleased to see Elijah go, and now they wish to see Elisha gone as well.

So these youths were no innocent group of children. They were a large gang of 40+ men, who surrounded Elisha mocking and probably threatening him. More importantly, they demonstrated their unbelieving and blasphemous hearts in the way they treated the anointed prophet of God. Therefore, Elisha acted under the influence of the Spirit to speak a curse against these men and their wicked ways.

Fulfilling Elisha's prophecy, the Lord sent two bears to attack the men. Elisha walked away unharmed, yet the forty-two men did not. Also note the Bible says the youth were torn apart, which means gravely harmed but not necessarily killed (though some may have died). This was the Lord’s judgment on these men, not merely Elisha’s. Elisha didn’t control the actions of the bear, but we know God did.

In conclusion, this moment was orchestrated by the Lord to ensure the people of Israel understood that Elisha would have the same power God gave to Elijah in his day. In fact, Elisha performed twice as many miracles over his tenure as did Elijah.
 

Scripture quotations taken from the (NASB®) New American Standard Bible®, Copyright © 1995, 2020 by The Lockman Foundation. Used by permission. All rights reserved. www.lockman.org