Why does Bane protect the girl?

This is actually explained in the prisoner's story, as he tells it to Bruce:

The legend is that there was a mercenary who worked for the local
warlord. He fell in love with warlord's daughter. They were married in
secret.

When the warlord found out, the mercenary was condemned to this pit.
But then he exiled him instead. The mercenary understood that was the
daughter who had secured his release.

But what he could not know was the true price of his freedom. She took
his place in the pit. And she was with a child. The mercenary's child.

Innocence cannot flower underground. It was to be stamped out.

One day, the doctor forgot the lock the cell. But the child had a friend. A protector who showed the other that this innocence was their redemption, it was to be priced.

The mother was not so lucky.

This is Bane's prison now. He wouldn't want the story told.

Saving this child was the one good (actually, the only significant) thing that Bane could do in his life. He had no prospect of ever leaving the Pit or doing anything remotely significant.

The accepted answer mentions that Talia's mother may have told him about Ra's Al Ghul and that he might have expected to be rescued as a reward. I think this is deeply wrong and maybe even offensive to his motives. A little girl, who has never even seen the outside of the Pit, who had no resources whatsoever, whose father didn't even know that she existed,... her mere survival would be a hard gamble. Also, his own survival of the riot is a minor miracle in itself, and he must have been aware that he is not likely to survive helping Talia. So, expecting any kind of reward is wrong and I strongly believe that his action was completely pure and exclusively focused on the salvation of that one child.

At the end of the movie, when Talia says her goodbyes to Bane, he cries, as he knows it's the last he sees of her, and that both of them are to be dead soon. Obviously, he has huge empathy for her... and only for her.

PShin wrote that "Bane is like a second father to her". This is close, but I think still incorrect. Bane is the father to her (not biological, but in emotional, protective, and every other way) and he sees her as his true daughter. I assume he took her under his protection "as his redemption", and grew very fond of her through the years he cared for her in the Pit.

After all, it is not hard to see that an innocent child was his only (metaphorical) light in the long years he spent there. Hence, his contempt for all but her, and huge devotion to her (and only her). I also think that her feelings are quite reciprocal, and for the quite similar reasons.