To state the obvious one first: Just calculate the probabilities the straightforward way. This requires no advanced thinking and as long as you just slavishly follow the rules, you'll reach the correct conclusion.
If you don't follow that path, the point of all explanations is to counter the wrong intuition that the last step is an independent new choice.
One way is that if you chose from the beginning to keep your first choice, then of course the probability of winning cannot depend on whether the quiz master opens another door first (and we know in advance that behind that door there will be a goat). So the probability of winning when not changing is the probability of choosing right the first time, which is $1/3$. The probability of winning when changing therefore has to be $2/3$, because you know that the prize is behind one of the doors.
Another way is to note that if you have chosen the wrong door from the beginning, the show master actually has no choice of which door to open. Therefore he gives you the additional information: "If you have chosen wrong, that other door I didn't open contains the price." Again, the probability that you've chosen right is $1/3$, and therefore the probability that the other door contains the prize is $2/3$.
Also note that it is important that the quiz master always opening the empty door is part of the game. To see that, assume the other extreme: The stingy show master who opens the prize door whenever possible, in order to avoid the candidate winning the price. In that case, it's obvious to everyone that if he opens a goat door, the best strategy is not to change, because if one hadn't chosen right from the beginning, he certainly would have opened the prize door instead. This demonstrates clearly that the second choice is clearly affected by the strategy of the quiz master, and therefore argues against the notion that you have to have a probability of $1/2$ for each choice afterwards; of course that argument doesn't provide you with the actual probabilities of the original problem, but it at least makes it plausible that just like the quiz master favouring the prize door makes it more profitable not to change, the quiz master favouring goat doors makes it more profitable to change.