New Keynesian IS curve: question about time dependence
As you already mentioned this equation can be derived from the households consumption rule.

the equation is derived based on consumption C rather than income Y, by considering the marginal utility of comsuming right now vs saving and delaying consumption. Investment/net export is not a part of this model, so C=Y

So lets take a look at the Euler condition for the household (where $\pi$ represents inflation, $\beta$ the discount factor and $R_t$ the noninal interest rate)  and remember the market clearing condition $C=Y$:
$$\begin{align}C_t^\sigma = \beta^{-1}  E \left[\frac{\pi_{t+1}}{R_t} C_{t+1}^\sigma \right]\end{align}$$
You can directely see, that if the household expects future consumption to increase, current consumption will also increase (assuming no changes in inflation and interest rates). This is called consumption smoothing. The households tries to balance consumption over different periods. A rise in nominal interest rate makes saving more attractive and thus households shifts some of the current consumption into future periods.