Pg. 120
First half:
どうやって = "how; in what way; by what means
バグってる = to be buggy (てる is stative in this case)
の = stand in for a noun, being described by the preceding verb
が = subject marker
人間 = human
だ = copula
と = quoting particle, taking all the preceding things and summarizing for the following verb
分かって = to understand/to know, in て form to connect to the remainder of the sentence.
So, the first half: “How do (they) know the buggy thing is humans and…”
Second half:
バグが = bug, subject marker for second sentence
何日か = how many days
続くって = to continue, stative (so in a state of continuing)
発表 = announcement
できたん = can, in past form, with a shortened の
だろう = seeking confirmation, but pretty confidently. In conjunction with the shortened の from before, のだろう is used, mostly by men, to basically confirm info based on something said earlier (in this case, based on Kasane telling them the number of days)
So, “They are able to announce how many days the bug will continue.”
Overall, with some shuffling around for better English, I take this to mean: “How do they know that the buggy thing is humans and are able to announce how many days it will continue?”
It all kinda precedes the next panel, where he points out that whether we are observing the world or humans, or should be the same. Implication being that the way we observe the world, for all intents and purposes, is the world. He’s basically questioning how the agency works and how they can determine both that the world isn’t flipped (just human cognition) and how long it goes for.
At least, that is how it reads to me.
If there are any specific grammar questions I’m missing, feel free to ask. I’m typing this in a hurry before I get ready for work, so I’m not as detailed as I might usually be!