How is property divided when a mother dies without a will and has three kids?

Since your sister died

Since your sister died without a will to probate, Missouri's intestate laws will determine who inherits your sister's home. When a person dies without a will, they die "intestate."

Inheritance descent depends on who the survivors are. You don't mention that your sister has a surviving spouse. If she did have a surviving spouse, the spouse would be entitled to a part of the property, shared with your sister's sons. If there is no surviving spouse, then your sister's sons are entitled to the entire intestate property (including the house), to be equally divided between them.