Textbook claims that price level accompanied by decrease in interest rates?
Presumably the context is deriving the AD curve from the IS-LM model. Start from money market equilibrium $$M/P=L(i,Y).$$
If $P$ falls then the supply of real money balances rises. With $M$ and $Y$ fixed the nominal interest rate $i$ must fall in order to induce people to hold the greater supply of real money balances. This is because the alternative to holding money is holding it in bonds which pay interest. To get people to hold more money (which pays no interest) we need the opportunity cost of holding money to be lower, so yes your interpretation is correct. 

With sticky inflation expectations (recall the Fisher equation $i=r+\pi^e$) the change in the nominal interest rate will map one to one to changes in the real interest rate. This means that the LM curve shifts down at any given level of $Y$ in $(r,Y)$ space. This produces a new equilibrium in the IS-LM model with a lower real interest rate and a higher income (since the lower real interest rate has stimulated investment). Thus a negative relationship between $P$ and $Y$: the AD curve.