An otherwise unknown person (or group of people) calling himself (or themselves) "Satoshi Nakamoto" released a paper discussing their ideas on how a decentralized crypto-currency might work. Then they release the source code for a computer program that would implement their idea.
Several talented and knowledgeable computer programmers reviewed the computer program and agreed that it accomplished what "Satoshi" said it did. They chose to try running it and exchange the resulting "bitcoins" mostly for entertainment and proof that the concept worked. As more people discovered the software and verified its safety and security, it became more popular. Eventually it was popular enough and people saw that there was value in what it could accomplish. This intrinsic value became a starting point for exchanging between bitcoin and other currencies.
Here is that original paper if you'd like to read it yourself. It's quite interesting:
http://bitcoin.org/bitcoin.pdf