As JavaLatte alluded to, and as Canadian Yankee said more clearly in a comment, Kasky is describing the feeling of looking into the barrel of a gun which is being pointed at his face. The preposition "down" is used to describe how he is "looking." I don't know that this makes any logical sense; it is just an idiom that must be learned.
And again, as JavaLatte alluded to, "look down the barrel of a gun" often has the meaning of "aim down the barrel of a gun" or "sight down the barrel of a gun"—that is, to be the person holding the gun and aiming it. When describing the action of being-pointed-at, it is more common to use "stare down the barrel of a gun." But Kasky is using a parallel construction—he is looking at Rubio and looking down the barrel of a gun.
Now, I disagree with JavaLatte that Kasky's meaning is unclear. It is this:
Kasky was a survivor of the shooting attack carried out by Nicholas Cruz. In that attack he may have, at one point, literally "looked down the barrel" of the gun Cruz was using or he may not have; I don't know the details. But he was there at the school at the time.
He is now on stage with Senator Marco Rubio, who is a gun-rights advocate and accepts campaign donations from the pro-gun National Rifle Association.
In describing his feelings on sharing the stage with Rubio, Kasky says it is hard to look at Rubio and not feel like he is looking at the person who attacked his school; it is hard to look at Rubio and not feel like he is "looking down the barrel of an AR-15"—that is, it is difficult for him to not feel like he is being threatened, because the Senator will not enact policies that prevent these shootings.