How are 1x1 convolutions the same as a fully connected layer?
Your Example

In your example we have 3 input and 2 output units. To apply convolutions, think of those units having shape: [1,1,3] and [1,1,2], respectively. In CNN terms, we have 3 input and 2 output feature maps, each having spatial dimensions 1 x 1. 

Applying an n x n convolution to a layer with k feature maps, requires you to have a kernel of shape [n,n,k]. Hence the kernel of you 1x1 convolutions have shape [1, 1, 3]. You need 2 of those kernels (or filters) to produce the 2 output feature maps. Please Note: $1 \times 1$ convolutions really are  $1 \times 1 \times \text{number of channels of the input}$ convolutions. The last one is only rarely mentioned.

Indeed if you choose as kernels and bias:
$$
\begin{align}
w_1 &amp;= 
\begin{pmatrix}
 0 &amp; 1 &amp; 1\\
\end{pmatrix} \in \mathbb{R}^{3}\\
w_2 &amp;= 
\begin{pmatrix}
2 &amp; 3 &amp; 5\\
\end{pmatrix} \in \mathbb{R}^{3}\\
b &amp;= \begin{pmatrix}8\\ 13\end{pmatrix} \in \mathbb{R}^2
\end{align}
$$

The conv-layer will then compute $f(x) = ReLU\left(\begin{pmatrix}w_1 \cdot x\\ w_2 \cdot x\end{pmatrix} + \begin{pmatrix}b_1\\ b_2\end{pmatrix}\right)$ with $x \in \mathbb{R}^3$. 

Transformation in real Code

For a real-life example, also have a look at my vgg-fcn implementation. The Code provided in this file takes the VGG weights, but transforms every fully-connected layer into a convolutional layers. The resulting network yields the same output as vgg when applied to input image of shape [244,244,3]. (When applying both networks without padding).  

The transformed convolutional layers are introduced in the function _fc_layer (line 145). They have kernel size 7x7 for FC6 (which is maximal, as pool5 of VGG outputs a feature map of shape [7,7, 512]. Layer FC7 and FC8 are implemented as 1x1 convolution.

"Full Connection Table"

I am not 100% sure, but he might refer to a filter/kernel which has the same dimension as the input feature map. In both cases (Code and your Example) the spatial dimensions are maximal in the sense, that the spatial dimension of the filter is the same as the spatial dimension as the input.