For the speaker in "Birches," swinging on birch trees symbolizes a temporary return to a youthful carefree state.
The narrator is reminiscing about his past - when he was a young boy, without a care in the world, who would spend his days swinging on birches and enjoying his happy youthful life. Now, when he is an old man and cannot swing on those trees anymore, he misses his childhood when everything was much easier.