There's five reasons intertwined to explain the perception that she's responsible.
She has a personal reason to hate Klingons.
She killed a Klingon first. Without the Federation side knowing what the
Klingons thought about it, she could be blamed for instigating the initial standoff to begin with, as the video proof that she acted in self-defense was explicitly stated to have not been recovered.
She mutinied. That's the problem Starfleet officially has with her, but it's the thing that makes her well-known.
She killed the Klingon leader on the scene, in the process getting Georgiou killed, and creating the unifying martyr Burnham herself had worried about.
The Federation and Starfleet were unaware that nothing would have prevented T'Kuvma from instigating a battle.
Officially, her charges revolved around the mutiny, not the initial killing or Georgiou's death, but that gets wrapped into the perception she's responsible for the whole thing. Federation civilians would blame her for starting the whole mess. Starfleet personnel, with more context, blame her for being their first mutineer. Her former crewmates blame her for the captain's death. And everyone blames her for killing T'Kuvma.
So you've got someone known to have a reason to hate Klingons, who was the first to kill a Klingon, who advocated attacking the Klingons without provocation, who attempted mutiny in order to carry out that attack, who was with her captain when the latter was killed attempting a plan Burnham came up with, and who killed the leader whose death the Klingons rallied around, and they don't know that the Klingons would have attacked even if Burnham hadn't done all those things. Of course people would consider her responsible. And since Burnham doesn't know the last fact either, that she had been in a no-win situation where there would be no good outcome, she'd hold herself responsible as well.