Let's Play started in Late 2006, with a Something Awfulmember named Slowbeef. It first started even earlier, with his walkthroughs even earlier, then turned in to threads of screenshot walkthroughs, like the books you'd buy in stores, except with the creator's own commentary. Something which would be a mix of informative, witty, and humorous on a more crude or obvious level.
In 2007, Let's Play grew further, in to a Video Style. Something which quickly caught on. With such a method, the hobby caught on even faster, and on to various video hosting sites. Most commonly, you see this on YouTube, with the right search. Something Awful had created a subforum for this exact hobby.
What is Let's Playing, though? Do you really want to know?
It's not quite a walkthrough- Calling it that would assume that you don't have any commentary what so ever, and it just helps people get through the game. No personal flavour, no style that is unique to yourself. Whether it's through video or screenshot, Let's Play interacts with the viewer, the game, and themselves. How exactly is different, but hopefully the Let's Play Archive is a good enough set of examples for you.