I know that the phrase, "I'm going out on a limb here" means either to take a risk or hazard a guess, but where does it come from? As in, what did it originally refer to before it became an idiom?
The first uses of it in a figurative sense, with no reference to actual trees or climbing, come from the USA at the end of the 19th century. For example, the Steubenville Daily Herald, October 1895:
[...] If we get the 14 votes of Hamilton we've got 'em out on a limb. All we've got to do then is shake it or saw it off.
It is likely to be about the risk of climbing onto a branch of a tree.
This states that it was used by Steubenville Daily Herald in 1895:
"We can carry the legislature like hanging out a washing. The heft of the fight will be in Hamilton country. If we get the 14 votes of Hamilton we've got 'em out on a limb. All we've got to do then is shake it or saw it off."
The branches of a tree become progressively thinner and weaker as you proceed further outward from the center of the tree's main trunk. Thus, the further you proceed "out on a [tree] limb" the greater the danger you put yourself in.
It's why a cat who pursues a squirrel to the outer branches of a tree puts itself (and its meal) at great risk: the outer branches are able to support the weight of the squirrel alone but not the combined weight of squirrel-and-cat, and both may fall.
The same would apply, for instance, to a person chasing a cat or a leopard chasing a monkey.
I'm going to go out on a limb here, and totally dismiss the reference from the Herald above as being not in the real sense as I know it. This seems to indicate a series of actions designed to move an opponent to a position whereby they are trapped and can be killed of - a sort of "give em enough rope" expression. My understanding of out on a limb, is that others see a branch and believe it to be too weak to hold any sort of weight. I, however, by being an expert on tree mechanics totally believe it is possible to walk on it to pick the apple. Despite being told and warned, I go out on the limb and heyho, I pick the fruit. I have trusted my instict, gone against all popular thinking and backed a seemingly dangerous and costly action, but I know without doubt is the correct course. I would say this term comes from a tree, or possibly maritme, with beams and sail arms in a storm.
from the bible.... Zachius, the tax collector, wanted to see Jesus as he was entering the town but he was too short so he ran ahead of the crowd and climbed a tree, out on a limb. Just about that time jesus walked under and said "Zachius come down, I need a place to stay tonight." Zachius came down and was so moved by Jesus that he offered half of what he owned to the poor. So, Zachius took a chance, and was rewarded by Jesus' presence and the spirit of generosity.