Playing 1/2 NL in a private game

(I'm relatively new to the site, so I may not have all the information that regular posters usually include)

Hero is dealt Ah 3h on the button.

SB - $100
BB - $300
Hero - $175

There is a call from mid position, Hero calls, SB calls, BB checks.

Flop: Kh Kc 6h

SB bets $4

BB raises to $12

MP folds

Hero calls

SB calls

Turn: 7d

SB bets all in $86

BB raises $161 (raising Hero all in)

Results in white below:

Hero folds

(BB takes back $75, uncalled)

River: 4d

SB shows Ks 6s (Kings full of sixes)

BB shows Kd 7h (Kings full of sevens)

BB wins pot of $216

I'm fairly new here and am wondering what everyone can say about this hand? Should I have folded on the flop? What other kind of approaches would you people take with this hand? Any input is appreciated.