He broke his oath by making conflicting promises. The Nahel-bond breaks when Kaladin strains it trying to save his life
Kaladin gave conflicting oaths: to protect Elhokar and to help Moash. Giving conflicting oaths is apparently something that hurts the Nahel bond, at least when it forms with honorpren.
She scrunched up her face. “I don’t want to force you to do anything. You
have to do what you think is right.”
“That’s what I’m trying to do!”
“No. I don’t think you are.”
“Fine. I’ll tell Moash and his friends that I’m out, that I’m not going to
help them.”
“But you gave Moash your word!”
“I gave my word to Dalinar too. . . .”
She drew her lips to a line, meeting his eyes.
“That’s the problem, isn’t it,” Kaladin whispered. “I’ve made two
promises, and I can’t keep my word to both.” Oh, storms. Was this the sort of
thing that had destroyed the Knights Radiant?
What happened to your honorspren when you confronted them with a
choice like this? A broken vow either way.
Words of Radiance
The bond gets weaker in an insidious way, with Syl acting more inattentive and childlike. What happened when Kaladin falls into the chasm is that he tries drawing stormlight. That's when he hears Syl scream in his head, like that act was too much of a strain in the weakened bond.
As the bridge fell out from beneath him, Kaladin reached for Stormlight.
Nothing.
Panic surged through him. His stomach dropped and he tumbled into the
air.
The fall into the darkness of the chasm was a brief moment, but also an
eternity. He caught a glimpse of Shallan and several men in blue uniforms
falling and flailing in terror.
Like a drowning man struggling toward the surface, Kaladin thrashed for
the Stormlight. He would not die this way! The sky was his! The winds were
his. The chasms were his.
He would not!
Syl screamed, a terrified, painful sound that vibrated Kaladin’s very
bones. In that moment, he got a breath of Stormlight, life itself.
Words of Radiance
It's only after this that Kaladin doesn't find Syl.
If only he could find Syl. Everything would be better if he could see her
again, if he could know that she was all right. That scream . . .
Words of Radiance