Coincidentally, I had just watched
this Numberphile video on the Monty Hall problem (there are many to choose from!) that makes the answer much more intuitive.
The solution is not clear when there are just three doors so instead, say that there are 100 doors. You pick a door. It's clear that your odds of having picked the correct door are 1 in 100. Now Monty Hall eliminates 98 of the other 99 doors, leaving only one other door remaining. It's easier to see in this example that you would obviously want to switch to the one door of the 99 that Monty did not eliminate. The odds of that door being correct are 99 in 100.
The same holds true for 3 doors but in this case the odds of your initial pick are 1 in 3 and the odds of the other non-eliminated door are 2 in 3.