Here's how the plot actually works in SchT (Schooltale), if anyone's interested:
How the monsters escaped the Underground in Schooltale:
Long ago, after the great war, humans imprisoned the monsters under Mount Ebott using the Barrier. Only the power of seven human souls could destroy it. One day, a human fell into the Underground. They were wounded and dying. Toriel and Asgore wanted to help them, but unfortunately they couldn't save them.
Asgore took their soul and went to the surface. He was the first monster in centuries to set foot on the land of humans. Using the power of the soul and his own royal magic, he destroyed the Barrier from the outside.
Meanwhile, down below in the Underground, Toriel and Gaster were organizing the exit. Gaster, then still a young scientist, worked to keep the passage stable while monsters ascended to the surface. He helped Toriel lead the monsters, coordinated, made sure no one was left behind. It was then that he became friends with Asgore, and this friendship became one of the most important in his life.
The monsters founded the city of Ebott and began a new life. And Gaster, three years after the liberation, created Sans through his experiments, and two years later, he created Papyrus!
How the humans became part of the Dreemurr family and how Asriel was born:
After the liberation, Asgore and Toriel dedicated themselves to the city. Asgore remained the king of monsters and became the school principal, while Toriel became the vice principal and his right hand. They dreamed of a family, but for a long time they couldn't have children. Monsters already had more difficulty reproducing than humans, and after living in the Underground, the magic of many had weakened. Toriel had almost given up hope.
One day, while passing through a neighboring human city on business, Toriel and Asgore found themselves near an old orphanage. Toriel noticed three human children in one of the rooms. They were huddled together, fast asleep. These were Frisk, Chara, and Kris. They were about 4, 2, and 7 years old.
Toriel couldn't walk past them. They took all three. The paperwork took several months, but Toriel came every day, bringing books and sweets. She simply sat with them, reading aloud. The Dreemurrs hadn't planned to adopt three. They had planned on one. But Toriel said they couldn't be separated. And so Frisk, Chara, and Kris became Dreemurrs.
A couple of years later, a miracle happened: Toriel became pregnant. It was unexpected, joyful, and terrifying all at once. When Asriel was born, a little white goat with soft fur and big eyes, the whole family gathered around him. Frisk, who was about six at the time, smiled so widely that Toriel cried.
Asriel grew up surrounded by love. He had three older siblings who were ready to die for him, and parents who loved all four of them equally, never making a distinction between adopted and biological. :)