Explain how and why the cancellation of $ 6 $’s in $ \dfrac{16}{64} $ to get $ \dfrac{1}{4} $ is a fallacious statement. Based on what we know from elementary and middle school teachers, most of us know that 16/64 correctly equals 1/4 because 16/64 is simplified with a common divisibility of 16. However, there is another way to prove that 16/64 equals 1/4 without dividing the numerator and denominator by 16. Who can explain how and why that method leads to a fallacious statement?

It's not fallacious.  $\frac{16}{64}$ is equal to $\frac 14$.
But the method is unsound, because it only works for a few special fractions.   For example, if you try it with $\frac{12}{24}$  you get $$\require{cancel}
\frac{12}{24} = \frac{1\cancel 2}{\cancel 2 4} = \frac14$$ which is completely wrong. 
