The speaker prefers to think the birches have been bent by boys instead of ice storms because, first of all, boys do not do permanent damage to the trees to the extent that ice storms do. In lines 4 and 5, the speaker says,
"...swinging doesn't bend them down to stay
Icestorms do that."
Even more than this, however, the speaker likes to think boys have bent the birches because he used to do so himself, and he recalls what a valuable experience it is for a young boy to be able to climb to the very top of a birch and let himself down. Being able to engage in such an activity can provide a lonely boy an outlet for his energies, and allow him to keep himself interested and occupied in the absence of friends. In lines 24-27, the speaker says,
"I should prefer to have some boy bend them
As he went out and in to fetch the cows -
Some boy too far from town to learn baseball,
Whose only play was what he found himself."
Although the speaker does not directly state it, it is implied that, when he was a boy, he was isolated from the company of others his own age, and found companionship and play in the birches instead.
In the extended metaphor in lines 41-49, the speaker compares the action of the boy's play with the birches to a wider concept of life. The boy climbs the birch tree, and when he is at the top, he feels separated from the earth, high above it. By shifting his weight, he can get the tree to slowly bend and bring him back to earth when he is ready. The speaker likens this experience to being able to escape from earth temporarily, when he would
"...like to get away from earth awhile
And then come back to it and begin over" (lines 48-49).
Climbing to the top of the birches offers one the chance to "get away from it all" for a bit, allowing for a period of rest and renewal before getting on with the sometimes difficult business of life.