Textbook claims that price level accompanied by decrease in interest rates? I read from this textbook the following, quoted as:


  In addition, a decline in the price level is usually accompanied by a
  fall in the rate of interest...


I interpret this as meaning: "A fall in price causes interest rates to fall".
This quote, in context, was meant to explain the interest rate effect - a reason why the aggregate demand curve slopes downwards.
Can someone verify and explain why this is the case? Additionally, is my interpretation correct?
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.