Isn't it the photograph that is flipped? After all if I'm in front of a mirror my left hand appears to the left and the right hand appears to the right. But on a photograph it's the opposite.

Intuitively just consider the difference between the two:

What DO you see? In case of the mirror, you see what's BEHIND you, because the stuff in front of you is blocked by the mirror, no? And just to be clear, by "you" I mean your eyes, and in particular the direction your eyes are looking at.

Now in contrast a photograph never shows what's BEHIND the camera (now that would be kind of funky, don't you think?), but - just like your eyes - only captures what's in front of the lens.

And there's your difference.

I don't know how to "explain" intuition, but usually I imagine not myself but some kind of geometric shape in front of a mirror (say a square with the edges labelled). If I try to visualize how light bounces, it is intuitive that in a mirror I'm seeing the light bouncing from mirror -> me in the proper way and in a photo, I'm looking at the shape directly with a camera (I'm where the mirror was).