Can someone help me understand how this works?
「」=Is this a quotation mark?
I can't understand what modifies what.
You really don't want to take writing cues from J-CAST “News”, as it is not exactly a high-quality news source…
In this case, most of the brackets are seemingly used to highlight keywords, for the clickbait factor. I would say that the only warranted use is 「レッドブル」, because Red Bull is a foreign brand name.
A saner rendition would be something like:
The brackets around 死亡報道 can maybe be justified too, as a form of “sarcastic quotes”, because the report turned out to be false. In fact, you might even interpret all of the brackets to be sarcastic quotes, because it's “very funny” that this trivial nonsense is masquerading as a “news article”.
Quotation marks are common even in English to quote tiny paragraphs of text. It often comes off as sarcastic, but usually it's intended to be a single word (or more) quote to validate what's being said.
Take the example from the BBC world news headline today:
"Mali hotel attack: 'No more hostages' after special forces raid".
It's not that "no more hostages" is sarcastic, but just a validation through quotes that it's been confirmed by someone else.
In this case, 「謝罪」validates that 謝罪 was actually said by Redbull, and isn't just the headline inventing a story.
「死亡報道」shows that the name 死亡報道 is unofficial, and just a name given to the event. Much like saying "the so called death reports" (いわゆる「死亡報道」). Or, you could argue that this is a quote as we see above with 謝罪; it's not totally clear what's being intended.
Finally, 「デマ情報」and「レッドブル」have quotation marks because they're the names of a news source and foreign company -- two organization names. It's non-standard Japanese (I don't think any language actually has this as standard usage), but whoever wrote the title had it in mind as a name. Some newspapers in English do this too. Again, from the BBC:
Oregon shooting and the anonymous '4chan' message board
And a Japanese example:
4chan just happens to be an easy example to find.
If talking from a standardized viewpoint, none of these usages are strictly formal Japanese, though I would argue that none of them are wrong per-say (I don't actually know what academics on 国語 have to say on this usage). In fact, as I've made some effort to point out, there's a lot of convergence on how this news source is using 「」and how western news organizations use quotation marks in their headlines and articles.