Sorta ELI5: You're both right and wrong. There are a couple different kinds of color blindness. The kind you mention wouldn't be helped by the glasses. The most common cause of red/green color blindness (anomalous trichromacy) is different though. All three kinds of cones are there, but there's too much overlap between M- and L-cones.

Normally M-cones like "green" light more and L-cones like "red" light more; both might fire for red light, but the brain knows that if L-cones are firing more than M-cones it's seeing something red.

With anomalous trichromacy, both cell types like "red" and "green" light the same, so the brain is confused since both cell types are firing the same amount. However it can still distinguish light on the extreme ends of "red" and "green" where the two cells don't overlap as much in sensitivity.

The glasses work by filtering the light; it makes "green" light more extremely green and "red" light more extremely red. (This is really an over-simplification. The graphs and explanation on the company's site might help.) They don't change the biology, but change the light so that more "green" light falls in the range that the weird M-cones like most and more "red" light falls in the range the weird L-cones like most.

Take-aways: If you have the more rare kind of color-blindness where you're missing a photoreceptor type, the glasses won't work. Also, it's sort of un-testable if the glasses make things look the same as a normal person; you CAN test whether people can then distinguish red and green though with a red/green colorblindness test. I don't know if that's been done for real, but the company's website claims they've done it. And I trust them, especially given the emotional reaction videos like this one which seem pretty honest (not advertising like the really famous vid), and the fact that the science isn't THAT crazy. Science is always pretty crazy though...