a song of ice and fire - In Game of Thrones season 2, why do they (possible spoilers)?

This was something that was changed from the book. My answer will be based on the book A Clash of Kings.

Initially Qhorin states that it will be easier for Jon if the rest do not watch.

“Then you must do what needs be done,” Qhorin Halfhand said. “You are the blood of Winterfell and a man of the Night’s Watch.” He looked at the others. “Come, brothers. Leave him to it. It will go easier for him if we do not watch.”

However it is later revealed that Qhorin didn't need her dead, he just
wanted to test whether Jon would kill her.

Qhorin’s shrewd grey eyes seemed to see right through him. “So you let her go?” He did not sound the least surprised.

“You know?”

“Now. Tell me why you spared her.”

It was hard to put into words. “My father never used a headsman. He
said he owed it to men he killed to look into their eyes and hear
their last words. And when I looked into Ygritte’s eyes, I...” Jon
stared down at his hands helplessly. “I know she was an enemy, but
there was no evil in her.”

“No more than in the other two.”

“It was their lives or ours,” Jon said. “If they had seen us, if they
had sounded that horn...”

“The wildlings would hunt us down and slay us, true enough.”

“Stonesnake has the horn now, though, and we took Ygritte’s knife and
axe. She’s behind us, afoot, unarmed...”

“And not like to be a threat,” Qhorin agreed. “If I had needed her
dead, I would have left her with Ebben, or done the thing myself.”

“Then why did you command it of me?”

“I did not command it. I told you to do what needed to be done, and
left you to decide what that would be.” Qhorin stood and slid his
longsword back into its scabbard. “When I want a mountain scaled, I
call on Stonesnake. Should I need to put an arrow through the eye of
some foe across a windy battlefield, I summon Squire Dalbridge. Ebben
can make any man give up his secrets. To lead men you must know them,
Jon Snow. I know more of you now than I did this morning.”

“And if I had slain her?” asked Jon.

“She would be dead, and I would know you better than I had before. But enough talk. You ought be sleeping. We have leagues to go, and dangers to face. You will need your strength.”

I've bolded the important parts of the quote.