It appears that there is no explanation because the 64 in "Graham's number" doesn't come from anywhere! The 64 doesn't appear in Graham and Rothschild's original paper on the topic, “Ramsey's theorem for $n$-parameter sets”; instead the paper has (p.290):

We introduce a calibration function $F(m,n)$ with which me may compare our estimate of $N^*$. This is defined recursively as follows:
$$\begin{align}
F(1,n)=2^n \qquad F(m,2)=4 &\qquad m\ge 1, n\ge 2, \\
F(m,n) = F(m-1, F(m,n-1)) & \qquad m\ge2, n\ge 3.
\end{align} $$
...

The best estimate we obtain this way is roughly
$$N^* \le F(F(F(F(F(F(F(12,3),3),3),3),3),3),3).$$

and according to this post by John Baez:

I asked Graham. And the answer was interesting. He said he made up Graham's number when talking to Martin Gardner! Why? Because it was simpler to explain than his actual upper bound - and bigger, so it's still an upper bound!