"Open" means "unfenced."
One of the core conflicts for the early settlers was between those who needed fenced ranches and those who needed open range. There were a number of these "range wars" in the West.
Range war - Wikipedia, the free encyclopedia
Feuds and Range Wars in the American West
Category:Range wars in the United States - Wikipedia, the free encyclopedia
Poetically, the term "open range" symbolizes the personal freedom of unfettered life, unrestricted movement, self-directed acts without social constraint.
Home, home on the range
Where the deer and the antelope play
~ Cowboy song
YouTube - Home On The Range - Gene Autry
There's no particular implication of "shooting" -- except of course, men living in the open range would have shot game for food. In addition, before land was annexed as part of the United States, there were no laws and no law enforcement, so shooting other people to settle disputes or to commit another crime (such as robbery) was perfectly possible.
The expression "open range" is not related to expressions such as "open season." This term does involve shooting game without restriction, but it is used in opposition to "closed season" -- the time of year in which it is illegal to shoot a specific type of game.
Open season (hunting) - Wikipedia, the free encyclopedia
In the expression "open range," "open" is the opposite of "fenced," not "closed."