I also know nothing but here's something missing from the blog post: "A random subgraph of the bunkbed graph is then formed by independently deleting each edge

I also know nothing but here's something missing from the blog post: "A random subgraph of the bunkbed graph is then formed by independently deleting each edge based on the assigned probability." So the (apparently incorrect) intuition is that an (upper<->lower) connection starts with an extra edge in the connection, so an (upper<->lower) connection has a greater risk of disconnect via random edge removal. Therefore, a same-level connection is more likely to remain after the random edge removal.