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...

This has more to do with place value.
Consider the image:

The key is the fact that $3^2 = 9$


*

*All the value below $3^2$ ($3^0 $ and $ 3^1 $) has to be accomodated in $9^0$
that is 

*all the values in $9^0$ have to be accommodated in the two places below $3^2$ ($3^0 $ and $ 3^1 $)
Similarly for any other places.



*

*This rule also applies to conversion between binary and octal, or binary and hexadecimal, or conversion between any base $k$ and base $k^n$


In general, for conversion of a number $N$ in any base $k$ to base $k^n$, each digit of $N_{k^n}$ get converted to n digits of $N_k$
