Ammunition consists of bullets, cartridges, or shells that are loaded into a gun. A person with a gun must stop killing when they run out of ammunition.
Consider a person, in a business perhaps, who is trying hard to get the organization to adopt some policy. Figuratively, that person is fighting a battle, and uses facts as his "ammunition."
"I dont want to give him any more ammunition" means "I don't want to do anything that he can use as facts to support his case."
Here is a possible example of use:
"You don't need to get to work until nine. It's only seven now. Why are you leaving so early?"
"I think the boss wants to fire me. She is looking for reasons. I was late twice this month because of traffic. She complained about it. I don't want to give her any more ammunition."