It is from this video. It is at 2 minute and 56 second. The contex is a coach eplains how to correctly do dumbbell press.
A lot of times you can cast your thumbs because you are trying to get a little extra cheat.
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If you straighten your right arm in front of you and make a fist (so that if you were to hold a pole, it would be perpendicular to the floor), then tilt your fist forward, notice that your thumb tilts forward. This is what he describes as casting your thumb. This is the motion he's trying to describe when he raises the dumbbells to do the exercises. He tries to explain that there are three ways to position your wrist/grip:
I wouldn't say this is a common use of the verb cast, but it's understandable in this context. It's roughly meant as follows (M-W):
a : to cause to move or send forth by throwing • cast a fishing lure • cast dice
The best that I can deduce is the coach simply misspoke. From the subject matter/context, I believe he meant to say:
A lot of times you can sprain your thumbs because you are trying to get a little extra cheat.
Indeed the words sprain and cast are related. A sprain is a type of injury where one or more ligaments are damaged. When one sustains a sprain, particularly a thumb sprain, a common treatment is having the affected body part restrained in a cast until it heals. In other words, getting your thumb placed in a cast is what happens after you sprain it. And so, it is to be unsurprising that someone who is simultaneously lifting weights while trying to teach might unwittingly mix-up the two words.
So to conclude, it would seem the coach was just trying to say that you shouldn't use your thumbs as conveyors of force while doing dumbbell presses, instead, your hand should remain in a neutral grip to avoid potential injuries.