Understanding Tiernan's Algorithm
I think I have managed to figure it out from the paper, and would like to leave my explanation for the next person who struggles with this concept.

From my understanding, Tiernan means nothing of a mathematical nature when he says "$u$ is closed to $v$". He's actually trying to say "$u$ is closed off to $v$", i.e. no longer accessible.

By following along with Tiernan's paper and his example, his algorithm manages to discover a cycle with the following vertices:


  $v_1, v_2, v_4, v_3, v_5$


Now that he has discovered a cycle, he removes the last vertex $v_5$ to see whether another vertex, say $v_x$, can be added to the remaining nodes, e.g. $v_1,v_2,v_4,v_3,v_x$, in the hopes of finding another cycle. 

However, to avoid rediscovering the same cycle (such that $v_x \ne v_5$), the vertex $v_5$ is now closed to $v_3$ and $v_3$ cannot extend an arc to it again. In the matrix $H$, we now have $H[3,1] = v_5$ (or simply $5$ since Tiernan requires unique integer labels for each vertex). If $v_3$ would become closed to another vertex before $v_3$ was removed from $P$ (to backtrace and attempt to discover cycles using $v_1, v_2, v_4, v_x$), then that vertex would be placed in $H[3,2]$.

In a nutshell, "closed to" should actually be "closed off to/inaccesible" due to the fact that the node was already used as the last node in either a cycle, or a "non-cyclic dead end". This prevents the algorithm from rediscovering the same cycles over and over.