Well, something has to receive the action of the transitive verb (or one in which the action of the verb must be transmitted to an object) 'to watch'; that something must be a noun. Gerunds are verbs masquerading as nouns, and they name an action. So, those which are receiving the action of watching are the actions of weaving and dancing. This could be reworded as "...to watch the Native Americans weave and dance." This would eliminate the necessity of the gerund by supplying another noun for the transitive verb (Native Americans are now being watched; the action itself is not now being observed).
I hope that was helpful.