I'm not sure how. This isn't saying that we're all connected via quantum energy and then we can harness that energy via shrooms.
That's kinda exactly what it's saying.
Quantum entanglement just doesn't work like that, it doesn't travel faster than light, the entangled particles have to have become entangled together then be separated - not be part of the same species or organism - and it can't be used to communicate information FTL. Also entanglement is just some BS pulled out of the arse of whoever made this video. From what we've had from the show so far it looks like a case of "Quantum" ergo "magic".
Also as of the 3rd episode I'm 50-50 on whether they are saying this Quantum magic extends to all living things..
The nice thing about inertial dampeners and the rest was that it just supposed physics we don't know about yet.
Not that old Trek was above having a plot line like intragalactic space mushrooms on some of its sillier episodes, but the combination of the grimdark tone and magic space mushrooms has me reaching for my vomit bag.