Put oneself in an isolated or disadvantaged position in one's support of someone or something.

The clear allusion in this phrase is to climbing trees. All of us must remember that feeling of not wanting to go further out to reach that apple/ball or whatever for fear that the branch (limb) would break under us. 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:

"We can carry the legislature like hanging out a washing. The heft [main part] 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."