I don't know if the important thing is gestures per se so much as it is a sense of the theatrical.
Actors are unique in the performing arts because they "play" two instruments simultaneously. The first and most obvious one is their voice. But the second is their body -- how they move, carry themselves, and relate physically to the other people on stage. A great actor can take the same words another actor just read and turn them into a completely different character, just by the way they use their body.
When people think about public speaking they mostly focus on the voice, but just like in acting the body is important too. The great speakers know how to use the voice and the body together to reinforce the messages each sends to the audience.
If you want to develop your skills with voice and body, acting classes can be a great way to do that, as can training in improv.
Body language and gesture is also important in a wide variety of musical performances. Especially since music, rhythm, dance, and body movement are all so deeply intertwined. Perhaps actors and opera singers (which would also include actors in musicals... sing, act, AND dance) might manifest these traits more obviously, but body language is important to nearly all human performance.
As a musician (not a singer), I communicate with other musicians largely via body language, but the audience is largely oblivious to it. Only another musician -- and mostly just those who have played with me -- would understand the difference between lifting my bow 1 cm or 2 cm off the strings of my violin.
This is true, there certainly are many cases where the audience is oblivious. But imagine the stereotypical rock/metal band playing. Their stance, how the strut around, how they move, how their hair moves all effect how the audience reacts to the performance.
Actors are unique in the performing arts because they "play" two instruments simultaneously. The first and most obvious one is their voice. But the second is their body -- how they move, carry themselves, and relate physically to the other people on stage. A great actor can take the same words another actor just read and turn them into a completely different character, just by the way they use their body.
When people think about public speaking they mostly focus on the voice, but just like in acting the body is important too. The great speakers know how to use the voice and the body together to reinforce the messages each sends to the audience.
If you want to develop your skills with voice and body, acting classes can be a great way to do that, as can training in improv.