While we have a strong flush draw, betting into a paired board on the turn and calling a raise leaves us in a spot where we have zero showdown value on the river.
Checking back is preferred to realize our equity. When BB checks, their range is often protected with Ax or mid-pairs that won't fold to a single small bet.
Note: Betting is acceptable but checking is higher EV; we have high equity but low fold equity against a 3-bettor's range on an Ace-high board.
Checking is the standard play here. The board pairing the 9c is better for the BB's range, and our hand has excellent equity to just take a free card.
Note: Betting into a paired board against a range that contains many 9x and Ax hands is unnecessary; checking back ensures we see a river.
Once we bet and get raised, we are priced in to call. We have a nut flush draw and the pot odds are too good to fold, even if we are occasionally drawing dead to a full house.
The river is a brick and we have nothing but King-high. Even with excellent pot odds, we cannot call a bet because we beat absolutely nothing in the BB's value or bluff range.