First, everybody is a little different biomechanically, and I'm not about to argue with Brandon Curry that doing what he is talking about has helped to build his biceps.
But, from a anatomy standpoint the biceps brachii performs 3 joint actions/movements:
1) Supination- turning the palm "up" (or towards the body when performing a curl). This is why doing curls with the palms up or towards you makes regular curling variations better for your biceps than reverse curls or hammer curls where your palms/wrists are pronated (down/away from you) or neutral (facing towards each other/thumbs up).
2) Elbow flexion- Bending the elbow joint
3) Shoulder flexion- raising the upper arms up in front of the body. While the biceps are not the prime movers in this movement (as opposed to the other two movements), they are an important synergist (assisting muscle). Obviously variations in hand/wrist position will make this movement more or less biceps intensive (palms down takes the biceps mostly out of the equation, palms up hits them much better).
So, actually from a biomechanical/anatomy standpoint raising the elbows slightly will involve the biceps more. The problem with free weights is that the amount of resistance experienced by the muscles is dependent upon gravity and therefore leverage. The closer to vertical the forearms/lower arms become, the greater the leverage and therefore the less resistance from gravity the muscles experience. That's what Brandon was talking about in that segment.
The thing is, unless you've got biceps the size of Brandon's, you're probably going to be able to get the bar up to neck/mouth level without having to raise your elbows so high that your forearms go completely vertical. Also, doing them by lifting the bar up to mouth/neck level hasn't seemed to hurt Skip, or these guys either:
If you feel your version better in your biceps then go for it. I always felt curls like that more in my brachialis. To each his/her own I guess.