Obviously (?) eliminating one door increases the odds that your first choice was correct from 1 in 3 to 1 in 2.

I think that this is where you may be getting confused. As you say, in your initial choice of door, the odds are 1 in 3. At this point, you know that at least one of the other doors has a goat behind it.

When the host (who knows where the car is) opens one of the other doors to reveal a goat, he is merely confirming what you already know: one of the other doors has a goat behind it. Therefore, and this the counter intuitive part, your odds are still 1 in 3. Nothing else has changed.

Hence switching increases your odds of winning to 1 in 2 at this point (or 2 in 3 overall).