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ITT are a bunch of bad explanations; some are ok but here's more detail:
Imagine a transparent sheet of glass with words printed on it. Hold it up so you can read it. Now stand in front of a mirror. You can read the words, both in the mirror and on the glass in front of you. Neither set of words is flipped.
Now imagine turning the glass around so the words face the mirror. As you rotate the glass 180 degrees horizontally, it flips around mirror image. Both in the glass in front of you, and in the mirror, you can now see a "mirror image" view of the text. But the mirror didn't do anything -- you did, when you chose to flip the glass around horizontally instead of vertically to face the mirror. If you'd tumbled it top over bottom to face the mirror, then the text wouldn't have flipped left and right, but it would have flipped high and low.
Your head is similar to the text, except you can't see through it. But, if you rotated yourself to face away from the mirror, and your head was transparent, someone looking at you would see a not-mirror-image face of yours, both looking at them and in the mirror from the other side. Same as the text. Nothing is flipped. When you chose to spin yourself horizontally to face the mirror, instead of doing a headstand, you chose the transformation that will led to seeing the mirror image instead of the high-and-low flipped image of your face. If you face away from the mirror, and do a handstand instead of spinning horizontally, you'll see a vertically flipped face with its left and right preserved.
Hopefully that makes sense.