Why can we convert a base $9$ number to a base $3$ number by simply converting each base $9$ digit into two base $3$ digits? 
Why can we convert a base $9$ number to a base $3$ number by simply
  converting each base $9$ digit into two base $3$ digits ?
For example $813_9$ can be converted directly to base $3$ by noting 
\begin{array} \space 8_9&=22_3 \\ \space 1_9 &=01_3 \\ \space 3_9
 &=10_3 \\ \end{array}
Putting the base digits together ,we get $$813_9=220110_3$$

I know it has to do with the fact that $9=3^2$ but I am not able to understand this all by this simple fact...

Consider $N$ in base 3. For simplicity, we can assume that $N_3$ has an even number of digits: if it doesn't, just tack on a leftmost $0$. So let:
$$N_3 = t_{2n+1} t_{2n}\dotsc t_{2k+1} t_{2k} \dotsc t_1 t_0.
$$
What this positional notation really means is that:
$$
N = \sum_{i = 0}^{2n+1} t_i 3^i,
$$
which we can rewrite as:
$$\begin{align}
N &= \sum_{k = 0}^{n} (t_{2k+1} 3^{2k+1} + t_{2k} 3^{2k}) \\
&= \sum_{k = 0}^{n} (3 t_{2k+1} + t_{2k}) 3^{2k} \\
&= \sum_{k = 0}^{n} (3 t_{2k+1} + t_{2k}) 9^{k}. \\
\end{align}$$
But now, note that for each $k$, $3 t_{2k+1} + t_{2k}$ is precisely the base-9 digit corresponding to the consecutive pair of base-3 digits $t_{2k+1} t_{2k}$.
