Why can soldiers of an invading army commit atrocities without being bothered by them?
Because when people are removed from home and community, no one knows them, so the old Adam has free course. The restraining influence of surroundings, circumstances, parental guidance, and community is gone. In that strange environment, the old Adam acts unchecked, so invading soldiers can pillage and rape without it bothering them.
The speaker explains this by recalling basic training, where young recruits did things they never did in their hometowns. When asked why, they said, "Because nobody knows me." Even an unbeliever can be inhibited by surroundings, circumstances, parental influence, and community. But put someone in a totally strange environment, away from home where no one knows them, and the old Adam has free course. That is exactly what happens when soldiers of an invading army pillage a community and rape the women, and it seemingly does not bother them. It is the old Adam. Every person, before salvation, would have been capable of those same things, because that is where the human race is coming from. Paul is showing that the old Adam abounds with sin, but where sin abounds, the grace of God is always greater. The old Adam has reigned like a king over us individually and over humanity as a whole, but grace can also reign like a king.