Short answer: Yes. But this video is nonsense:
Research has proven that a mouthpiece in general can aid in fast-twitch explosive motions like throwing a baseball or swinging a baseball bat through various physiological responses. (The reduced release of cortisol is one such touted response.) However, it is completely unnecessary to purchase a $60 baseball mouthpiece from Under Armour to get this effect. You can spend $2 at the local sporting goods store and have a clear mouthpiece/mouthguard which will have the exact same effect and will actually protect your jaw as well.
Endorsing the UA mouthpiece is just an example of shilling a product to make money when there is a clear alternative that works just as well and isn’t insanely expensive.
Conclusion: Save your money. Go to the local sporting goods store and buy a $2 mouthpiece. Skip the stupidly expensive Under Armour garbage. (This is good advice in general, by the way.)