Why does deflation cause banks to increase their interest rates? I have read that in a deflationary spiral, lower prices cause banks to increase their interest rates. Is this true and what is the logic behind it?
I suspect there is a small mistake in your notes. 

Deflation does not cause banks to increase their interest rates. However it is true that a deflationary spiral (or plain deflation for that matter) causes real interest rates to increase. These are the interest rates that matter for the economy anyway, which is why in macroeconomics we often refer to the real interest rate even if it's not explicitly stated so.

The nominal interest rate, set by banks in part, is the interest rate in terms of money. It determines how much money we have to pay back for credit.
The real interest rate  is the rate in terms of goods and services. It determines how many goods and services we have to pay back for credit.

The real interest rate is defined as:

$ (1+r) = (1 + i) / (1 + \pi) $, which can be approximated by:
$r = i - \pi$, where r is the real interest rate, i is the nominal interest rate set by banks and $\pi$ is the inflation rate.

When we have deflation we get a negative $\pi$, which, as can be seen from the equations above, increases the real interest rate r. The smaller $\pi$ is, the bigger r gets. When $\pi$ is negative, even if we set $i$ to its lowest possible value, which is 0, the real interest paid on it will be positive. This means the amount paid back in goods and services is larger than the amount borrowed and increasingly so the bigger the deflation (smaller $\pi$) is.