I can't remember buying a game because of watching a Let's Play, but I do remember one time a Let's Play made me play a game I had already bought.
After hearing good things about the original Mount and Blade in 2008 I picked it up. I started the game, played through the "tutorial" which is pretty much just showing you how to control your character, and was totally lost. Didn't get it at all. So I put the game down and never came back to it.
Then a few years later, I think around 2012, I watched a Let's Play where someone had a big army of Nordish troops, was in a battle with them, giving them orders, and it started to click what the game was about. I watched some more of the series, then went back to play it.
Between the original and Warband I've got well over 500 hours into the series and have enjoyed it quite a lot. I've since used Let's Play to help me decide if I want to buy a game, but I've yet to not even know about a game, watch a Let's Play, and then buy it. I think this is because I rarely watch Let's Plays of games I don't own.