The author actually explains this in the excerpt you've provided:
(...) in order that we may use matrix multiplication rules.
If you multiply two $N\times N$ matrices $A$ and $B$ with components $a_{ij}$ and $b_{lk}$ together, you write: $$ C = AB = \sum_{m=1}^N a_{im}b_{mk} $$ So as you see, the $(i,k)$th element of the resulting matrix $C$ is the scalar product of the $i$th row of the first matrix with the $k$th column of the second matrix. To ensure this, the indices you sum over have to somehow be "next to each other" in the notation.
If you take your formula $$ \Lambda^{\mu'}{\;\;\mu}\Lambda^{\nu'}{\;\;\nu}\,\eta_{\mu'\nu'} = \eta_{\mu\nu} $$ and want to write it in matrix form, you see that after you change the multiplication order $$ \Lambda^{\mu'}{\;\;\mu}\Lambda^{\nu'}{\;\;\nu}\,\eta_{\mu'\nu'} = \Lambda^{\mu'}{\;\;\mu}\,\eta{\mu'\nu'}\Lambda^{\nu'}_{\;\;\nu} $$ the $\nu'$s are next to each other, but the $\mu'$s aren't. So you have to switch around the indices in the first $\Lambda$, which is done via the transpose.
Only after you did that you have the correct order of the indices and you can write the equation in matrix form.
Addendum: Let me try to explain the process in more detail in a simpler toy model. Although you say you do not understand how to go from tensor notation to matrix notation, this has actually nothing to do with the objects being tensors, so let's just drop the distinction between co- and contravariant indices and see if that way we can clear things up.
In the very simple example of two $2\times2$ matrices $A$ and $B$ $$ A=\left(\begin{matrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{matrix}\right) \qquad B =\left(\begin{matrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{matrix}\right) $$ when we start at calculating $A\cdot B$ it is clear what has to happen: $$ C:= A\cdot B = \begin{pmatrix}a_{11} & a_{12}\end{pmatrix}\cdot\begin{pmatrix}b_{11}\b_{21}\end{pmatrix}\cdot\left(\begin{matrix} 1 & 0 \ 0 & 0 \end{matrix}\right) + \begin{pmatrix}a_{21} & a_{22}\end{pmatrix}\cdot\begin{pmatrix}b_{11}\b_{21}\end{pmatrix}\cdot\left(\begin{matrix} 0 & 0 \ 1 & 0 \end{matrix}\right) + \begin{pmatrix}a_{11} & a_{12}\end{pmatrix}\cdot\begin{pmatrix}b_{12}\b_{22}\end{pmatrix}\cdot\left(\begin{matrix} 0 & 1 \ 0 & 0 \end{matrix}\right)+\begin{pmatrix}a_{21} & a_{22}\end{pmatrix}\cdot\begin{pmatrix}b_{12}\b_{22}\end{pmatrix}\cdot\left(\begin{matrix} 0 & 0 \ 0 & 1 \end{matrix}\right) = \left(\begin{matrix} a_{11}b_{11}+a_{12}b_{21} & a_{11}b_{12}+a_{12}b_{22} \ a_{21}b_{11}+a_{22}b_{21} & a_{21}b_{12}+a_{22}b_{22} \end{matrix}\right) $$ This is what I described above by "scalar product of row with column", but it should be clear from your linear algebra course how to do it. 945