It took me a long time to figure this out as a child, since nobody I asked knew the answer! Is it something about the vertical of the mirror being different than the horizontal? No, if you rotate the mirror 90 degrees it still inverts left/right and not top/bottom. Is it something about the Earth that makes horizontal different from vertical? No, if you look into a mirror while your body is in horizontal position, it still inverts left/right, and not top/bottom (viewed from your body). Is it something about our two eyes being on a horizontal line instead of a vertical line? No, if you close one eye it still works all the same. If you trace out how the light goes, you'll see that there is nothing different about horizontal vs vertical. Yet clearly a mirror inverts things horizontally but not vertically. How can this be?
The real answer is that we perceive this because we are used to turning in the horizontal plane. If you see yourself in the mirror and you raise your right hand, the guy in the mirror is raising his left hand. But how do you know that it is his left hand? Mentally you place yourself in his position by rotating yourself and then you check which hand it is, but by doing this rotation your left hand ends up in the position of your right hand in the mirror. So even though the alignment of you and your mirror image appears to be perfect on first sight, it actually isn't. If we were creatures that are used to only turning in the vertical plane, then we'd perceive the same image as top/bottom inversion. You'd mentally rotate yourself vertically, and then instead of your left hand ending up where you see your right hand and vice versa, your head would end up where you see your feet and vice versa.
The fact that our bodies are roughly symmetrical in one direction but not in the other makes this a bit non obvious unfortunately. It may be easier to understand this with an object that's not symmetrical in any direction. If you have a piece of paper with some word written on it, and you look at it via a mirror, why is the word flipped from left to right and not from top to bottom? This question has the same answer: because we're used to turning things in the horizontal plane. To show the paper to yourself via the mirror, you rotated the paper so that the text is now facing away from you, towards the mirror. But how did you rotate it? You rotated it by turning it in the horizontal plane. If you had rotated it in the vertical plane, the word would be flipped vertically. It's just that if you do this nobody is surprised that the word ends up flipped vertically, because that's not how our psychology works.
The real answer is that we perceive this because we are used to turning in the horizontal plane. If you see yourself in the mirror and you raise your right hand, the guy in the mirror is raising his left hand. But how do you know that it is his left hand? Mentally you place yourself in his position by rotating yourself and then you check which hand it is, but by doing this rotation your left hand ends up in the position of your right hand in the mirror. So even though the alignment of you and your mirror image appears to be perfect on first sight, it actually isn't. If we were creatures that are used to only turning in the vertical plane, then we'd perceive the same image as top/bottom inversion. You'd mentally rotate yourself vertically, and then instead of your left hand ending up where you see your right hand and vice versa, your head would end up where you see your feet and vice versa.
The fact that our bodies are roughly symmetrical in one direction but not in the other makes this a bit non obvious unfortunately. It may be easier to understand this with an object that's not symmetrical in any direction. If you have a piece of paper with some word written on it, and you look at it via a mirror, why is the word flipped from left to right and not from top to bottom? This question has the same answer: because we're used to turning things in the horizontal plane. To show the paper to yourself via the mirror, you rotated the paper so that the text is now facing away from you, towards the mirror. But how did you rotate it? You rotated it by turning it in the horizontal plane. If you had rotated it in the vertical plane, the word would be flipped vertically. It's just that if you do this nobody is surprised that the word ends up flipped vertically, because that's not how our psychology works.