The default method of communication if Voice is not enabled is typing in Second Life. I suppose in the interest of realism, some one programmed a typing animation into the mix, forcing your avatar to gesture wildly as he/she presses keys quickly on an invisible keyboard whenever he/she speaks.
It is a curious animation and many individuals dislike it. As with many options in Second Life, ridding yourself of it is not simple.
You need to access the 'Advanced' tab in order to do so, but that is not visible or accessible unless you 'enable' it by pressing the following keys on your real (and visible) keyboard:
Ctrl Alt D
Press the three keys simultaneously to bring up a new 'Advanced' tab in SL. You will find it at the top of your screen to the right of the 'Help' tab.
Now you need to enter the text that will allow you to disable the typing animation. To do this, go first to 'Debug Settings'.
You need to type 'PlayTypingAnim' in the box, but if you begin to type the word 'Play', often the rest of the text will appear as it is one of the most common settings that people change.
When it is entered in the box, you will see 'TRUE' is the default setting. Change that to 'FALSE' and you can close the box.
From this point, your avatar will be able to type in Second Life without being seen to do so!
N.B. If you are a dramatic sort of individual who likes his/her avatar to be in motion constantly but wish to change the actual mode of animation, you can purchase animations that cause your avatar to write in a book or scroll either with a contemporary pen or a feather quill... or any number of other alternatives. I personally find this a little overly dramatic but some avatars love it.