The way I try to explain it is in absolute (ish) coordinates:
If you face north, your image faces south. Your left hand is on your west side. The reflection of your left hand is also on the west side. If you grab the mirror and turn so you face west, your reflection faces east and both your left hand and the reflection are on the south side.
If you reset the mirror and then stand sideways (facing east, mirror to your north), your left hand is to the north and right hand is to the south. In the mirror, the image of your left hand is to the south and right hand, to the north. But your reflection is facing east, just like you are.
I think this is good too, but it leaves the reader with the original question "Okay, if the mirror doesn't reverse left-and-right, then why does it appear to do so?" A question which left me scratching my head for a while and frustrated that explanations given in otherwise excellent sources like Martin Gardner's The Ambidextrous Universe never actually addressed.
This is a good question and I think it's the perception that the person you see as a reflection is actually a clone turned around to face you. When you raise your left hand, your "clone" (if it is a clone) raises their right hand. If you imagine stepping through the mirror and turning around to stand next to your reflection you really will be reversed laterally compared to your reflection-twin.
If you face north, your image faces south. Your left hand is on your west side. The reflection of your left hand is also on the west side. If you grab the mirror and turn so you face west, your reflection faces east and both your left hand and the reflection are on the south side.
If you reset the mirror and then stand sideways (facing east, mirror to your north), your left hand is to the north and right hand is to the south. In the mirror, the image of your left hand is to the south and right hand, to the north. But your reflection is facing east, just like you are.