Even though Helaman’s forces had been tired from fleeing before the Lamanites, and then from marching through the night, they achieve their goal of arriving first to Manti. They then add themselves to the defense of the city. Even though tired, the Lamanites would not have known who they were, only that there was now a large army defending Manti. Thus, the Lamanites flee into the wilderness, apparently returning to Lamanite lands.
When Helaman says that they took possession of the city of Manti “without the shedding of blood,” he was being more figurative that realistic. Gid and Teomner certainly killed some Lamanites when they took the depleted forces that remained. Perhaps they did not suffer casualties, but that was unlikely.
There was a conflict between the Lamanites and the Nephites before the Nephites fled, and it is possible that more blood was shed. The figurative use of the idea was to note that it was a stratagem that allowed them to take the city without the massive loss of life it would have taken had they not been able to draw out the Lamanite army.