The Pri Megadim (siman 256) writes, המוסיף חיים לו מוסיפין הקודש על מחול, "Whoever adds from the weekday to the Shabbos, he will live a longer life."
We can explain it like this: Adam and Chavah ate from the Tree of Knowledge on Friday afternoon. The Midrash states that if Adam and Chavah had waited until Shabbos, they would be permitted to eat from the Tree of Knowledge. Their sin was that they ate from the tree too early when it was still Friday afternoon. For this aveirah, death came to the world. The Imrei Emes zt'l says that when we make tosfos Shabbos, we turn Friday afternoon into Shabbos. This means that, in retrospect, when Adam and Chavah ate from the Tree of Knowledge, it was already Shabbos. They ate the fruit at a time of day that people now consider Shabbos. Consequently, tosfos Shabbos atones for their sin. Their sin brought death to the world. By making tosfos Shabbos and rectifying their sin, it will save people from their deaths and enable them to live long.
Reb Simchah Kaplan zt'l (Rav of Tzfas) related that when he was young, he learned in Mir (Poland) and ate and slept in the home of a kind, elderly family who had only one son. He noticed that the parents were very scrupulous with tosfos Shabbos. When he asked his hostess why, she replied, "We have only one son. He was born twelve years after our chasunah. We were so happy when he was born, but the child wasn't healthy. The best doctor in Cracow, where we lived, told us that he had a heart condition and recommended we go to Vilna, home of a heart specialist – the top in the field.
"We took our son to Vilna, but even this great specialist couldn’t help us. 'There is no known cure for this condition,' he said.
"Before returning to Cracow, we stopped in Radin to speak to the Chofetz Chaim. The Chofetz Chaim was old and frail at the time, but Baruch Hashem, after making some hishtadlus, we got in. We told him about our child's illness and the doctor’s prognosis. He replied, 'Why did you come to me? I can't be מתים מחיה.'
"We broke down and cried.
"The Chofetz Chaim's grandson was in the room, and said to his grandfather, 'This is their only child; he was born to them after twelve years of waiting. And now the child's life is in danger. We can’t just send them away empty-handed.'
"The Chofetz Chaim asked us, 'Are you ready to be mekabel to keep tosfos Shabbos?'
"We replied that we were. The Chofetz Chaim said, 'Bring in Shabbos early, when it is still well before Shabbos, and you will see yeshuos.'
"As we were riding home, the child began feeling better. By the time we came home, our son was entirely well. We brought our son to the doctor. The doctor, an irreligious Yid, was shocked by the miracle the Chofetz Chaim performed. He gave us money and asked us to return to the doctor in Vilna to show him the miracle that had occurred to our son. We traveled to Vilna to ensure that our child was completely out of danger. When the doctor saw the child, he said, 'This can't be the child you brought to me before! The child I saw was beyond hope.'
"We told the doctor about the Chofetz Chaim's brachah and promise. The doctor said, 'No doctor in the world could’ve cured your son. The Chofetz Chaim made a miracle, מאין יש.'"
After telling this story, she told Reb Simchah Kaplan, "Now I'm sure you understand why we are so careful with tosfos Shabbos."
