What's the verb that describes "the act of using your hand to illustrate your words"?

Solution 1:

To gesticulate can be used to refer to that specific action:

• To make gestures especially while speaking, as for emphasis.

(AHD)

• to make or use gestures, esp. in an animated or excited manner with or instead of speech.

(Random House Kernerman Webster's College)

• Whenever she wished for anything very much she would gesticulate in a very expressive manner.

In your specific case to pantomime can be used:

• To represent or express by pantomime: pantomime a story on the stage; pantomimed "baby" by cradling an imaginary infant.

• He extended his arms and pantomimed (the action of ) water rushing out. (Also mimic would fit in the contex).

Solution 2:

It's not right for the context you've used, but I've seen people being referred to as a "hand-talker" when they excessively wave or gesticulate (with their hands) while speaking.

http://www.today.com/health/why-you-cant-stop-talking-your-hands-6C10526377