Jesse suggests that Winnie wait until she is older and then drink from the spring.
The Tucks have become immortal. They did not do this on purpose. They drank from a spring that caused immortality. Winnie discovered their secret, by accident. When Jesse Tuck realized that a young girl had seen him drink from the spring, he knew that the family had to tell her their secret so that she would not accidentally become immortal too.
The Tuck family kidnapped Winnie and told her the whole story of how they discovered they became immortal from drinking a spring in the wood outside her house. She realized that they were not scary people, they were just trying to keep her from making the same mistake they had.
Once you become immortal, you have to stop living a normal life. Everyone else sort of passes you by as they pass through life’s stages. For Miles, it was particularly sad.
"I was more'n forty by then," said Miles sadly. "I was married. I had two children. But, from the look of me, I was still twenty-two. My wife, she finally made up her mind I'd sold my soul to the Devil. She left me. She went away and she took the children with her." (Ch. 7)
As Miles’s story shows, it was impossible for Jesse and Miles to have normal lives, because their wives and children would grow old while they would not. If their children were to drink from the spring, they would remain children. As the Tucks explain to Winnie, she cannot drink from that spring, because it would have disastrous consequences.
"Do you understand, child? That water—it stops you right where you are. If you'd had a drink of it today, you'd stay a little girl forever. You'd never grow up, not ever." (Ch. 7)
As Winnie gets to know the family, she forms a bond with them and comes to get to know them all as people, but especially Jesse. He is closest to her in age, and really sees the potential in having a girl know about the spring. He makes her a proposal.
Look now—here's a bottle of water from the spring. You keep it. And then, no matter where you are, when you're seventeen, Winnie, you can drink it, and then come find us. We'll leave directions somehow. Winnie, please say you will!" (Ch. 22)
Jesse wants a chance at a normal life, with a family—or at least a mate. He sees in Winnie a partner. He likes her, and he knows that she likes him. He also knows that she is not quite old enough, and so he has to wait for her.
Winnie does make a choice, but she chooses to live a normal life. She ages, and dies. Why does she not choose immortality? We never know exactly. When Winnie pours the bottle on the frog, she knows she can get more. However, the stream is later destroyed. Winnie might have later met and married her true love. Maybe she decided not to be a teenager forever. Either way, Jesse understands.