Why is instantaneous utility of current period discounted? Consider a two period model of consumption.

I'm confused by the fact that in the optimum condition it is the marginal utility of the current period that is discounted, not the marginal utility of the next period.

Could someone give me an intuition behind this result?

Below, the derivation is presented.

$$ \max_{c_{t},c_{t+1}}u_{t}\left(c_{t},c_{t+1}\right)$$

such that 

$$c_{t}+\frac{1}{1+r_{t+1}}c_{t+1}=w_{t}+\frac{1}{1+r_{t+1}}w_{t+1}$$

FOC: (MU denotes marginal (aka instantaneous) utility)

$$\begin{cases}
MU_{t}=\lambda\\
MU_{t+1}=\lambda\frac{1}{1+r_{t+1}}\\
c_{t}+\frac{1}{1+r_{t+1}}c_{t+1}=w_{t}+\frac{1}{1+r_{t+1}}w_{t+1}
\end{cases}\Rightarrow\frac{1}{1+r_{t+1}}MU_{t}=MU_{t+1}$$
What you see as
$$
\frac{1}{1+r_{t+1}}MU_t = MU_{t+1} 
$$
can also be seen as
$$
\frac{1}{1+r_{t+1}}MU_t = \frac{1}{1}MU_{t+1} 
$$
where $1+r_{t+1}$ and $1$ are the respective prices of current and future money measured in future money. So basically what is happening is that you calculate how much utility an additional unit of future money would buy you in the current and in the future period. In optimum these quantities are equal. You can also rewrite the equation into the form of the familiar MRS condition:
$$
\frac{MU_t}{MU_{t+1} } = \frac{1}{\frac{1}{1+r_{t+1}}}.
$$