To go "out on a limb" means that you (1) put yourself at risk, (2) exert great effort, or (3) do something without support.
Most of us have two legs ... imagine going out with only one leg - "out on *A* limb." You can definitely see that this person on only one leg lacks support, must make a great deal of effort, or is in a precarious/unstable/dangerous situation.
(Meaning (1)): He hardly knew her, but he went out on a limb and asked for a date. (he put himself at risk of a rejection).
(Meaning (2)): Did you know that he went out on a limb for you and covered your missed assignments? (he exerted a lot of effort to cover the assignments)
(Meaning (3)): He went out on a limb and predicted who would win the race for mayor. (He had no support to make that prediction).