In Mankiw's Macroeconomics 7th edition, on page 99, there is the following equation, which states that "the price level depends not only on today’s money supply but also on the money supply expected in the future":

$$M/P=L(r+E_{\pi},Y)$$

where $L$ stands for the demand for real money balances and $r$ for the real interest rate. The author then poses the following thought experiment:

Suppose the Fed announces that it will increase the money supply in the future, but it does not change the money supply today. This announcement causes people to expect higher money growth and higher inflation. Through the Fisher effect, this increase in expected inflation raises the nominal interest rate. The higher nominal interest rate increases the cost of holding money and therefore reduces the demand for real money balances. Because the Fed has not changed the quantity of money available today, the reduced demand for real money balances leads to a higher price level.

My question is: I understand that, by the above equation, considering that $M$ is fixed, a decrease in the demand for real balances should lead to an increase in $P$. However, what exactly happens in the real world that causes the price level to rise as a result of people trying to get rid of the excess money they are holding? I can see that, if they were spending those "extra" balances on consumption, that would increase the demand for goods and services, which in turn would lead to higher prices. But since the reduction in $L$ was motivated by a rise in the interest rate, it is savings that is increasing, and not consumption. So, what is going on here? Thanks in advance.