Where does "Going out on a limb" come from?

Solution 1:

This one is actually quite straightforward. It alludes to going out on a branch of a tree. Etymonline says that the figurative sense is from 1897. The Phrase Finder supplies a quote from 1895:

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.

Solution 2:

The OED gives for "limb", sense 4: "A main branch of a tree".

The image is pretty obvious when you know this, I think.