Step 1: Picture the ADC as sorting into bins.
A flash ADC's job is to tell you which one of the possible output levels the input voltage falls into. If the converter has to distinguish 64 different output codes, imagine 64 boxes lined up side by side.
Step 2: Count the dividing walls, not the boxes.
A comparator is only needed at each boundary that separates one box from the next, because that is where the decision of whether the input is above or below a threshold actually has to be made. For $N$ boxes there are only $N-1$ walls between them, since the two end boxes only need one wall each on their inner side.
Step 3: Apply this to 64 channels.
Here $N = 64$, so the number of comparator thresholds, and hence comparators, is $64 - 1 = 63$.
\[ \boxed{63} \]