Context: Mizunashi Akari 水無灯里, a wannabe gondolier, gets nervous rowing a boat for the first time, rowing too fast until the passenger tells her to slow down.
kinchou de {ware wo wasure-sou ni} narimashita キンチョーで我を忘れそうになりました [I] became`{close to forgetting myself} due to nervousness. (literally.)
ware wo wasureru 我を忘れる To forget oneself. To be so focused in doing a task that you forget your own existence, in the sense of doing only one thing and forgetting about your surroundings (the passenger), or a greater objective.
