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The etymology of this expression is quite simple, being a simplification of an expression from a fable. Rather than spoil the fun, I will retell the fable here.
Note: this retelling is my own, based on my rough memory of the original fable. Please use and attribute accordingly.

Once upon a time, there were two merchants who had a camel. They needed to transport some goods to a nearby town. The first thing they loaded onto the camel was a bale of straw.

"Ah, my friend," said one merchant to the other, "we have all these things to sell, and it is so far away, and look at this camel, it is strong, it can certainly carry more."

The other merchant, who never spoke, nodded, and proceeded to load a collection of vases and bowls onto the camel's back.

"My friend," said the first merchant, "I still think the camel can surely hold more than this!"

The other merchant nodded, and loaded the vases and bowls with some of the grain they were going to sell as well.

For an hour, this continued; the first merchant would notice that the camel was strong, and could surely hold something more, and the second one would burden the beast with yet another item. Eventually, the camel was shaking under the load, but all of their goods were loaded on top.

The first merchant then noticed that when they had first loaded the bale of straw, however, one single piece had fallen out. "No sense in wasting even a single piece of straw. Friend, pick that up."

For the first time in ages, the second merchant spoke. "But the camel - it is quaking under the load. It will surely collapse if we load it up with anything more!"

Bewildered more by his associate's rebellion than by the fact he had spoken up, the first merchant frowned, and picked up the piece of straw on his own, and placed it on top of the load.

And it was the last straw which broke the camel's back.