Have you ever wondered why so many large chain stores have two sets of doors? No, it is not just to store shopping carts.
Where I live, any home that is around 100 years old (that hasn't seen any renovations) will very likely have two front doors. Putting it simply, you open one door, step into a small space, and there will be another door in front of you.
(Image Sources: Image 1 | Image 2)
The space goes by many names, including: arctic entry, mud room, breeze room, vestibule, airlock, foyer, and more. For sake of simplicity, I am going to call them "entry vestibules."
Entry vestibules create a buffer between the outside and the inside of the building, preventing drafts. This can help greatly with temperature regulation in both the winter cold and summer heat. I can't find number details on energy saving, but the fact that large chains still build them may hint of their importance for money-saving.
In homes, this space also typically serves as the mud room ( a place for shoes and jackets).
In the name of "first impressions," and open concept designs, vestibules are often the first thing to go during renovations, and I think that's a real shame.
It does two things: 1) a cache when one door is open, 2) and prevents drafts (from improper sealing around the door) like OP said. Insulation of the door is another component of the second part.