I like to think of a quotient thing as we keep everything but modify the equality on the given structure.
So, things like (some ring)$/(a-b)$ translate to [$a=b$ in the new structure].
Now, the original structure is $\Bbb Z[i]$, the Gaussian integers.
Then we modify the equality so that it respects the structure ($+,-,\cdot$), by posing $3=-i$.
In the new structure therefore every Gaussian integer $a+bi$ will be equal to the integer $a-3b$.
We get the integers $0,1,2,\dots,9$, and, as $3^2=(-i)^2$, we have $10=0$.
Now the only thing left to prove is that these ten numbers are still different in the quotient ring.