Think from the perspective of your first choice. You had a 66.66% chance of choosing the wrong door.
So now I just eliminate one of the doors you didn't choose. There are two left, including the one you chose first, which only had a 33.33% chance of being correct. Nothing else has changed about the problem.
What if we started with a thousand doors? You chose 1, with a 999/1000 chance of being wrong.
Of the remaining doors, I open 499 doors. The prize is behind one of the remaining 500 that you did not choose the first time. Do you want to have a chance to pick from one of the 500?
I like expanding it out to 1000 doors. Even though the logic is the same between 3 and 1000 doors, the chances are more intuitive with more doors.
I think the part that really throws the brain off with the 3 door version is that the host opening a door seems like another random event but it's actually dependent on the state of the game at that point. Our natural intuition of the chances is for a version of the game where you pick a door, then the host randomly picks (and does not open as you also haven't) a different door, and then you have to choose if you want to switch to the door that neither of you picked.
So now I just eliminate one of the doors you didn't choose. There are two left, including the one you chose first, which only had a 33.33% chance of being correct. Nothing else has changed about the problem.
What if we started with a thousand doors? You chose 1, with a 999/1000 chance of being wrong.
Of the remaining doors, I open 499 doors. The prize is behind one of the remaining 500 that you did not choose the first time. Do you want to have a chance to pick from one of the 500?