Where do bitcoins come from and what gives them their value?

Although individual bitcoins enter the Bitcoin economy as miners are rewarded for processing transactions, it's much more helpful to think of all 21 million bitcoins as having been created when Satoshi Nakamoto defined the Bitcoin protocol and launched the Bitcoin network in 2009.

The reason for this is that the Bitcoin protocol specifically defines and controls when and how a limited total number of coins are rewarded to miners for the job of securing the Bitcoin network. These "bitcoins" are really just mathematical tokens which are very carefully controlled by the network protocol to prevent counterfeiting, theft, etc.

By agreeing to use these mathematical tokens as money, the larger Bitcoin community is essentially "backing" their value and turning them into a currency in the same way traditional African and Asian societies used the money cowry despite the absence of any central bank. Unlike the money cowry:

there will never be more bitcoins
they are impossible to counterfeit
they can be divided into as small of pieces as you want
and they can be transferred instantly across great distances via a digital connection such as the internet.
Presumably, the members of the Bitcoin community who choose to accept them as money consider these features to be worth something, and value the bitcoin accordingly based on supply and demand on open currency exchanges.

Because bitcoins are given their value by the community, they don't need to be accepted by anyone else or backed by any authority to succeed. They are like a local currency except much, much more effective and local to the whole world.

So to sum up, bitcoins come from the Bitcoin community's agreement to follow a set protocol, and are backed by everyone who uses them as money.