We don’t know
Here’s the quote referenced in the question:
I released over a decade of pain and sorrow, a decade of loss. He held
me tight, and he smelled like my father, Epic or not.
“Son,” he said, clutching me, weeping. “I killed you. I didn’t mean
to. I tried to protect you, to save you. But you died. You died
It’s clear that this is not simply Blake feeling responsible for his son’s death: he actually killed him. Firefight confirms this:
“Look, I’ve told you before that you must meet him,” Firefight said.
“He refuses to believe what I’ve seen, but he needs to talk to you.”
“Why me? What does he care about me?”
“Well,” Firefight said, “he killed you.”
The book doesn’t elaborate on why this happened (perhaps saving it for a future series).
It’s possible that Blake killed David due to becoming an Epic and being infected by Invocation’s contempt for humanity (Invocation is the alternate-universe version of Calamity). This seems unlikely, though. People in the alternate universe talk as if the Epics there never felt compelled to evil:
And no Calamity,” I said, looking at the sky.
“The red star!” I said. “That brought the Epics.”
“Invocation?” he said. “It vanished a year after it arrived; it’s been
gone a decade.”
“Do you feel the darkness?” I demanded. “The drive toward selfishness
that strikes every Epic?”
“What are you talking about, Charleston?”
From what little information is given, it would seem that Calamity and its brethren were supposed to give humans powers and see what happened, not influence them in other ways (as Calamity did). So in the alternate universe, Invocation did its job properly and left when its time was up, and didn’t pollute humanity with its negative feelings.
Blake’s words fit with this, too: he probably wouldn’t have described himself as trying to “save” David if he had been trying to kill him out of malice. Perhaps it was an accident; perhaps he had to kill David for some reason or another, but had done everything to try to prevent it. Ultimately, at this point we just do not know how Blake caused his son’s death.
One highly speculative possibility is that the bank scene from Steelheart happened much as it did in the main universe, but with Blake Charleston as Steelheart instead of David. Recall that government agents came in and tried to subdue Steelheart, and their bullets bounced off of him. Perhaps, in the alternate universe, Blake tried to shield his son, but a bullet glanced off of him and killed David. But this is mere speculation, albeit consistent with Blake’s statements and the general similarities between the universes.