Is the Kalman filter actually forecasting? The state space equation is:
$$Y_t = F_tθ_t + v_t\hspace{4em} \textrm{eq. 1}$$
$$θ_t = G_tθ_{t-1} + w_t\hspace{2.8em} \textrm{eq. 2}$$  
$F_t$ in eq.1 are the independent variables and we can predict $Y_t$ provided we know $F_t$. 
Say that I am trying to forecast the number of hospitalizations due to flu based on humidity. In this case, based on eq.1 it looks like I need to know the humidity of next week to forecast the hospitalizations for next. In reality, I won't have any data about my independent variables for future too. I am actually not clear as to why this is called forecasting. Am I missing something here? 
 A: You are creating a model that has a dependent variable (Y) and an independent variable (X). By definition, you input an x and get a y. That's how it will work regardless of how (Kalman, ARIMA, LME, etc) you implement your model.
If you want to forecast hospitalizations without supplying next week's humidity, then you can use univariate methods (Kalman, ARIMA, etc), where you predict Y based on past Y only. If X actually is important, your univariate model will not be as good as one with X, but you have to decide whether it's too much trouble to get your X or not.
In practical terms, you could provide a range of X's, based on historic humidity readings, forecasts, etc, and then present a range of forecasted values. Or take the mean humidity for next week (over the last 20 years, say) and use that. Simply document how you got your X.
A: Your equations are right, and if you run the Kalman filter, your estimate of $\theta_t$ is indeed the filtered value of the state (the best, in least squares sense, value of $\theta_t$ given all information up to and including $t$).
If you want a prediction for the next time period with no additional information (which is what I think you are after), you can compute it as $\theta_{t+1|t} = G_{t+1}\theta_t$, and $Y_{t+1|t} = F_{t+1}\theta_{t+1|t}$; you would need a guess on $F_{t+1}$ if it is not available.
Hope this helps.
