Q:

Compute the pairwise Euclidean distance matrix

Can someone explain what is going on in the first two terms?
How does it compute pairwise Euclidean distance between rows in matrix $X$?
Is it $X^2 + (X^T)^2 -2XX^T$

In order to calculate the input pairwise similarity, we need to compute the pairwise Euclidean distance matrix $\mathbf D$ first. Using the matrix operations we could compute this matrix efficiently without using loops to do pairwise calculation:
  $$D = \begin{bmatrix} \vdots & \vdots & \vdots \\ \|\mathbf x_{\mathbf i}\|^2 & \ldots & \|\mathbf x_{\mathbf i}\|^2 \\ \vdots & \vdots & \vdots \end{bmatrix} + \begin{bmatrix} \ldots & \|\mathbf x_{\mathbf i}\|^2 & \ldots \\ \ldots & \vdots & \ldots \\ \ldots & \|\mathbf x_{\mathbf i}\|^2 & \ldots \end{bmatrix} -2 \mathbf X \cdot \mathbf X^{\mathbf T} \text{ where } \mathbf X = \begin{bmatrix} \vdots \\ \mathbf x_{\mathbf i} \\ \vdots \end{bmatrix}$$

A:

The first matrix is $\operatorname{diag}(X X^{\mathrm T}) \cdot \vec1$, where $\operatorname{diag}(X X^{\mathrm T})$ is a vector with the diagonal entries of $X X^{\mathrm T}$, and $\vec1$ is an all-ones matrix (with as much entries as $X$ has rows.) The second matrix is just the first one transposed, then.
For instance, if $X$ has three rows:
\begin{align}
X &= \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} 
\\ \\
X^{\mathrm T} &= \Big[ \matrix{ x_1^{\mathrm T} & x_2^{\mathrm T} & x_3^{\mathrm T} } \Big]
\\ \\
X X^{\mathrm T} &= \begin{bmatrix} \|x_1\|^2 & x_1 x_2^{\mathrm T} & x_1 x_3^{\mathrm T} \\ x_2 x_1^{\mathrm T} & \|x_2\|^2 & x_2 x_3^{\mathrm T} \\ x_3 x_1^{\mathrm T} & x_3 x_2^{\mathrm T} & \|x_3\|^2 \end{bmatrix} 
\\ \\
\operatorname{diag} (X X^{\mathrm T}) &= \begin{bmatrix} \|x_1\|^2 \\ \|x_2\|^2 \\ \|x_3\|^2 \end{bmatrix} 
\\ \\
\operatorname{diag} (X X^{\mathrm T}) \cdot \vec1 &= \begin{bmatrix} \|x_1\|^2 \\ \|x_2\|^2 \\ \|x_3\|^2 \end{bmatrix}  \begin{bmatrix} 1 & 1 & 1 \end{bmatrix}
\\
&= \begin{bmatrix} \|x_1\|^2 & \|x_1\|^2 & \|x_1\|^2 \\ \|x_2\|^2 & \|x_2\|^2 & \|x_2\|^2 \\ \|x_3\|^2 & \|x_3\|^2 & \|x_3\|^2 \end{bmatrix}.
\end{align}

