Actually, this idiom comes from another; "the straw that broke the camel's back". Even though a camel is a very strong animal, he does have a load limit. We think of straw as something that is very lightweight. However, if you pile straw on a camel's back, there is a limit to the amount he can hold. The straw that exceeds that limit is called, "the straw that broke the camel's back".
That philosophy has been applied to mental and emotional "loads" as well. For example, if a person is abused, there is a point at which he or she can take no more abuse. That point is called, "the straw that broke the camel's back".
Repeated usage of this phrase over the years has made it so common, that the phrase was shortened to, "the last straw". Now it has come into common usage to mean that a person has reached their limit of how much they can or will take concerning either abuse or work.