I mean from an organic chemistry perspective it's obviously true. If you throw an electron at a C=C double bond, you get a ·C-C¯: which will then go ahead and bond with an H⁺ that's floating around, leaving you with a radical still bound to your initial molecule, so the theory is

    CH₃-CH=CH-CH₃ + · + ROH ⇒ CH₃-ĊH-CH₂-CH₃ + :O¯R
    CH₃-ĊH-CH₂-CH₃ + HOR' ⇒ CH₃-CH₂-CH₂-CH₃ + ·OR'

and then the · on your ·OR' restarts the cycle, and the cycle doesn't stop until you get

    RO· + ·OR' ⇒ ROOR'

which is gonna be some random peroxide that wouldn't normally form, or

   CH₃-ĊH-CH₂-CH₃ + CH₃-ĊH-CH₂-CH₃ ⇒  CH₃-CH-CH₂-CH₃
                                          |
                                      CH₃-CH-CH₂-CH₃

which is gonna be some really random molecule with who knows what effects.

Now how important this is in a biological context I have no idea. There is a plausible mechanism for it to affect cellular chemistry, but the problem is that "a plausible mechanism" and "an actual effect" are very far from the same thing, and biochemistry is complicated.

The citation I'd want to see is that the chain reactions of free radicals do something biologically interesting, or, even better, that diets high and low in polyunsaturated fats lead to significant health differences in animal models.

If you don't mind, I'm going to use "nutrition grifter" moving forward.