she's been pondering it for awhile, the first realization was The Fortuneteller at the end when sokka makes a claim about how powerful aang is... Then they've sorta grown on each other, season 1 finale... season 2 episode one when she's talking about how scary it is to see aang in so much pain, it's ALL OVER the series, these are jus a few off the top of my head...
cave of 2 lovers is a big hint, serpent's pass when aang's talking to katara after the baby, she starts crying in joy, season 2 finale when she almost loses aang, season 3 episode 1 when she sorta flirts when him "I like your hair", when they dance and she's blushing, they grew on each other. Aang had the schoolboy crush right off the bat, but as he matured it turned into legit caring and love, and as Katara and aang were together more and more she started falling for him more and more...
Then you look at season 3 before the battle when they do the weird kiss thing, it was obviously Katara's first kiss (did you see anyone in the southern water tribe she woulda had a relationship with...)... And what I think personally, is Katara's been thinking about having a relationship with aang, but as stated in the ember island players she put her feelings on the back burner because of the war... Then after the final battle was all over and she had a chance to think it over, she decided to finally let her emotions run instead of holding them back so they wouldn't get in the way of what truly needed to be done