Why is the order of $\mathbb{Z}[i]/\langle 3+i\rangle$ equal to 10? The reasoning my book uses is as follows: 

Since $(3+i)(3-i) = 10$, $10 + \langle 3+i\rangle = 0 + \langle 3+i\rangle $. Also, $i + \langle 3+i\rangle = -3 + \langle 3+i\rangle = 7 + \langle 3+i\rangle$. So, $\mathbb{Z}[i]/\langle 3+i\rangle = \{k + \langle 3+i\rangle \mid k = 0,1,\ldots ,9\}$, since $1 + \langle3+i\rangle$ has additive order $10$. 

I do not understand why the additive order of $1 + \langle 3+i\rangle$ is $10$, or why the fact that $(3+i)(3-i) = 10$, shows the additive order or $1 + \langle 3+i\rangle$  is $10$. If someone could please explain this I would greatly appreciate it. 

Note: I apologize for not using MathJax to write this post. I intend to learn it over the coming weekend.
You might find clearer the following way of viewing the proof. Write $\,A = \Bbb Z[i]/(3+i)$.
Note $\, h\!: \Bbb Z \to A\,\color{#0a0}{ \ {\rm is\ surjective\  (onto)}}\,$ by $\bmod\, 3\!+\!i\!:\ \, i\equiv -3\,\Rightarrow\, a\!+\!bi\equiv a\!-\!3b\in\Bbb Z$
$\color{#c00}{I := \ker h = 10\,\Bbb Z}\ $ follows immediately by means of $\,\rm\color{#90F}{rationalizing}\,$ a denominator
$$ n\in I\iff 3\!+\!i\mid n\ \, {\rm in}\, \ \Bbb Z[i]\iff \dfrac{n}{3\!+\!i}\in \Bbb Z[i]\!\!\color{#90f}{\overset{\large \rm\ rat}\iff}\! \dfrac{n(3\!-\!i)}{10}\in\Bbb Z[i]\iff \color{#c00}{10\mid n}\ $$
Thus $\, \color{#0a0}{A = {\rm Im}\ h} \,\cong\, \Bbb Z/\color{#c00}{\ker h} \,=\, \Bbb Z/\color{#c00}{10\,\Bbb Z}\ $ by the First Isomorphism Theorem.
Remark $ $ Another view you might find illuminating arises from rewriting the ideal as a module in Hermite normal form: $\ I =
 (3\!+\!i) = (10,3\!+\!i) = 10\Bbb Z + (3\!+\!i)\Bbb Z.\,$ But it is trivial to test module membership given such a triangularized basis, namely
$$\begin{align} a\!+\!bi = a\!-\!3b +b(3\!+\!i)&amp;\in I = 10\Bbb Z + (3\!+\!i)\Bbb Z\\
\iff\ a\!-\!3b&amp;\in I\\
\iff\ a\!-\!3b &amp;\in 10\Bbb Z \iff  10\mid a\!-\!3b
\end{align}$$
Further this shows that $\, a\!+\!bi\bmod I\, =\, a\!-\!3b\bmod 10.\ $
The criterion generalizes to an ideal test for modules $\rm\,[a,b\!+\!c\:\!\omega]\,$ in the ring of integers of a quadratic number field, e.g. see section 2.3 Franz Lemmermeyer's notes linked here..
This is a special case of module normal forms that generalize to higher degree number fields, e.g. see the discussion on Hermite and Smith normal forms in Henri Cohen's $ $ A Course in Computational Number Theory.
Prop 1) $10 \equiv 0 + &lt;3+i&gt;$.

Pf:  $a \equiv b + &lt;3+i&gt;$ means there is a $k \in \mathbb Z[i]$ so that $a = b + k(3+i)$.

$10 = 0 + (3-i)(3+i)$ so $10 \equiv 0 + &lt;3_i&gt;$.

Prop 2) $i \equiv -3 + &lt;3+i&gt;$

Pf:  $i = -3 + (3+i)$ so $i \equiv -3 +&lt;3+i&gt;$.

Prop 3) If $a = j + mi; j,m \in \mathbb Z$ Then $a \equiv j - 3m + &lt;3+i&gt;$.

proof: $j+mi = j - 3m + m(3 + i)$.

Prop 4) If $a = j+ mi$ then $a \equiv r + &lt;3+i&gt;$ for some $0 \le r &lt; 10$.

Pf: We can find some $v$ so that $j-3m = 10*v + r$ where $0 \le r &lt; 10$.  That is basic division.

So  $a \equiv j - 3m  \equiv r + 10v + &lt;3 + i&gt;\equiv r + &lt;3+i&gt;$. 

ANd that's more or less it.  Any element of $\mathbb Z[i]$ is equivalent to an integer between 0 and 9.  So the set of equivalence classe $\mathbb Z[i]/&lt;3 + i&gt; = \{k + &lt;3+i&gt;| k = 0...9\}$

The only thing to show is if $j,k \in \{0...9\}$ and $j\ne k$ then $j \not \equiv k + &lt; 3+i&gt;$.