The reader knows that Jerry will see things differently now for a few reasons. The first reason is that Jerry set himself a goal. That goal was to swim through the tunnel. He trained for it. He prepared himself physically and mentally for it, and he succeeded in attaining that goal. He proved to himself that he could do it. Achieving an all consuming goal like that usually creates changes in people. For Jerry, his journey through the tunnel represents a rite of passage. He passed his rite of passage test, and is now no longer a boy.
The reader also knows that Jerry sees things differently now based on the last sentence of the story.
It was no longer of the least importance to go to the bay.
For four pages, that tunnel is a singular purpose for Jerry. It's all consuming, and it's the only thing that he can think about and work toward. After achieving that goal, it's not even a blip on his radar. He has moved on.
I like the way that you asked your question. You asked how the reader knows that Jerry will see things differently. Lessing takes a very physical approach to telling the reader that Jerry will see things differently. What body part sees? The eyes. Lessing tells the reader that Jerry's eyes have gone through a physical transformation during his rite of passage swim.
He could see nothing but a red-veined, clotted dark. . . his eyes were glazed-looking.
His eyes were strained from the swim and exertion, but it's important to note that Lessing emphasizes how the swim affected his eyes. She doesn't spend time talking about his lungs, or chest, or arms, or head. She tells the reader about his eyes, and how his eyes have been changed.