The quote in question appears at the very end of chapter five. A lot happens during this chapter. It starts with Winnie Foster testing out her idea of running away. She goes off to explore the woods near her house, and she is startled to see a boy drinking from a small spring. The boy is Jesse Tuck. Jesse and Winnie engage in some small talk, and Winnie eventually asks Jesse if the spring water is good to drink. Jesse is immediately on guard and tries to explain that Winnie should not drink from the spring.
"Believe me, Winnie Foster," said Jesse, "it would be terrible for you if you drank any of this water. Just terrible. I can't let you."
Winnie is persistent, though, and she keeps trying to take a drink. Jesse knows he's in real trouble in the current situation. He simply doesn't know what else to do.
It's at this moment that Mae and Miles Tuck show up at the spring. Jesse is relieved they are there because they can help deter Winnie. Mae quickly takes in the situation and realizes what has happened.
And at once, when she saw the two of them, Jesse with his foot on the pile of pebbles and Winnie on her knees beside him, she seemed to understand. Her hand flew to her bosom, grasping at the old brooch that fastened her shawl, and her face went bleak. "Well, boys," she said, "here it is. The worst is happening at last."
What Mae Tuck means is that her family's worst fear has just happened. Somebody else has discovered the spring. If it were a normal spring, that wouldn't be a big deal, but the spring grants the person who drinks from it immortality. The Tucks know the spring is both a blessing and a curse, and they do not want knowledge of it spreading around.