Local variance derivation by Gatheral

I've bought Gatheral's book on Local Volatility and I have troubles with understanding a part where he shows that local variance is a conditional expectation of instantaneous variance. Why in the second equation from the bottom he just skips the term $\theta (S_T-K)dS_T$? He says that it's because $F_{t,T}$ is a martingale. I see that $F_{t,T}$ is a martingale, but don't know how this helps. Also , what "condiational expectations" is he talking about? The notation looks a bit sloppy. Thanks for