I was able to hear both "Green Needle" and "Brainstorm", whichever I concentrated on.
Then I thought about testing it with both "Greenstorm" and "Brain Needle". I was able to consistently hear the former, but never the latter.
Then I opened the Time Magazine article with its link to the original Tik Tok, and suddenly wasn't able to hear anything but "Brainstorm" - until I paused repeating the clip for some minutes, which probably allowed my brain to reset. ;)
The thing is, our brains are very imperfect, messy little machines. We see images in clouds, or a white triangle pointing down in an image like this (https://upload.wikimedia.org/wikipedia/commons/5/55/Kanizsa_triangle.svg) not because they are really there, but because our brains need to cope with incomplete information all the time and thus have become good at completing anything into a "full picture", even if there isn't one in the first place.
Similar to optical illusions, the reason for auditory illusions is the fact that any input signal is already heavily interpreted before we even start to reason about it consciously. (By the way, that's one of the reasons why about 90% of an uncompressed song can just be thrown out, leading to comparably small MP3 file sizes.)
I guess this specific illusion works because the start of both phrases ("Green", "Brain") is very similar, so if there's a noisy sound like "[incomprehensible] [r] [something between ee and ai] [n]", your brain is able to understand either depending on context. By concentrating one one or another word, you're actually forcing that context, so that your biased brain just seeks confirmation and ignores the rest as noise.