For such a popular technology, how bitcoin started is still somewhat of a mystery. We don't really know when bitcoin's first design work was done, as the entire project was a secret and there are few details about how it may have come to be before 2009.
In 2009 the domain bitcoin.org was registered, and later that year a link on the site promoted a whitepaper by a person calling themselves Satoshi Nakamoto. The paper was titled Bitcoin: A Peer-to-Peer Electronic Cash System, and this is the first public information about bitcoin that exists.
Satoshi Nakamoto also mined the first block on the bitcoin blockchain, and before they (we don't know who they are) dropped out of view, it is estimated that Nakamoto mined more than 1 million bitcoin (worth around $15 billion at the moment).
We do know that before leaving the public eye, Nakamoto entrusted developer Gavin Andresen with far reaching responsibilities. Andresen became the bitcoin lead developer for the Bitcoin Foundation, which was more or less a group that helped bitcoin get off the ground.
There are few other details about the early days of bitcoin, and of course, there is a lot of speculation. It is possible that as time goes on, more information will become public, or Nakamoto will try and cash in one of the most valuable bitcoin hordes out there.