It is usually used to mean that something almost didn't happen, but it did. Examples: "I could barely lift the box" would mean that I almost could not lift it but was able to. "I barely made it to my appointment in time" means that I was almost late to my appointment but ended up making it on time. "I could barely understand him" means that I almost didn't understand him but was able to. "With the heavy trailer, the car could barely make it up the hill" means the car almost didn't make it, but ended up making it. Hope this helps!