I fail to comprehend the part in bold. The Invisible Man, Wells:
He vanished behind a laburnum, and appeared again clambering a fence that abutted on the open down.
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"Abutted" means the following:
(of a building or an area of land) be next to or have a common boundary with
So it means the fence is next to something.
The meaning of "open down" is:
An open down would be a green hill that had no fences. Some downs have been fenced - often with low stone walls, so sheep can graze on them. (wordreference)
So it means the guy was clambering a fence that was next to a green hill.
to be adjacent; touch or join at the edge or border (often followed by "on", "upon", or "against"):
This piece of land abuts on a street. Dictionary
"down" in this sense means hill, and is unrelated to the direction "down" (it is related to words like "dune"). It is quite specific to southern England in fact. Similar hills in the North are called "wolds"/"moors". "Down" is a word like "alp", used for the hills in a particular place, and not for hills in general.
Hat tip to @James K