Actually, Finny did not swing from a line. He jumped into the river. The passage refers to the first time during the Summer Session that Finny and the others challenged the tree that plays such a crucial part in the story. The tree, as it lived in Gene's memory, was "tremendous, an irate, steely black steeple beside the river." Because of its height and positioning by the river, the tree had become a part of the physical training being emphasized for Devon's seniors whom Gene describes as "draft-bait, practically soldiers, [who] rushed ahead of us toward the war." No Upper Middler (junior) at Devon had ever jumped from the tree until Finny made it their mission.
During the opening scene of the novel's flashback, Finny scrambles up the wood pegs nailed to the side of the tree and steps out on the one branch that reaches farthest toward the river. He asks if it is the branch the seniors jump from in their training, but gets no answer. Then he challenges Gene and the others: "If I do it, you're all going to do it, aren't you?" Again, getting no answer, Finny proceeds:
"Well," he cried out, "here's my contribution to the war effort!" and he sprang out, fell through the tops of some lower branches, and smashed into the water.
Finny was being facetious, having fun on this school day in summer with his friends. The war, at that point, still seemed very far away from them, but they were all aware of its presence out there somewhere, waiting for them when their turn came.
I agree with mshurn's answer, and I would like to add that it is also foreshadowing. The jump from the tree used for physical training will be Finny's only contribution to the war effort as he is turned down for military service in the future and then dies an untimely death before he can serve his country.