The phrase "insure domestic tranquility" means that the government should provide peace and order in the society. From the word itself, domestic - family, home, national and tranquility - state of being tranquil, calm, peace. The prase technically means the government should ensure the nation's peace and order.
I remember that she was interested in how the world would be later on? But I am not for certain. I have a copy of the book but It's just been a while since I read it.
