I understood how to see if a relation $R$ is antisymmetric, but how can I understand it from these tables? For example:
With $X = \{\,a, b\,\}$
$$\begin{array}{|c|c|c|} \hline & a & b \\\hline a & 1 & 1 \\\hline b & 0 & 1 \\\hline \end{array}$$
$R$ IS antisymmetric.
With $X = \{\,a, b, c\,\}$
$$\begin{array}{|c|c|c|c|} \hline & a & b & c \\\hline a & 1 & 0 & 1 \\\hline b & 1 & 0 & 0 \\\hline c & 1 & 0 & 1 \\\hline \end{array}$$
$R$ is NOT antisymmetric.
Why? What do I have to look at to understand the process?
If $R_{i,j}$ is the entry in row $i$, column $j$ of the table of relation $R$, the table of an antisymmetric relation has no occurrences of $R_{i,j} = R_{j,i} = 1$ whenever $i \neq j$. It's nothing but the definition of antisymmetry.
In your second example, $R_{1,3} = R_{3,1} = 1$, which says that $a$ is related to $c$ and vice versa; yet $a$ is distinct from $c$. Hence $R$ is not antisymmetric.