I started listening to gaming podcasts back in 2012 with Nerd Poker, as a way to pass the time during my 2hr a day commute, and then Critical Hit (a 4e podcast with an amazing DM, Rodrigo Lopez, and an fantastic and funny cast of small town Kansas theatre nerds). I hadn't actually played D&D in about 7 years, and hadn't played regularly since the 90s, but listening was a fun way to pass the time and remember the game I loved. But Critical Hit was so good that it gave me the itch to start playing again. When the free Next playtest material started coming out, I downloaded it, convinced my wife and a couple of friends to try it out, and now have been playing 5e 2-3 times a month for 6 years and introduced over two dozen first time players to the game.
I'd heard the hype about Critical Role, but like others, didn't have the time to sit down and watch 3-4 hour episodes. When they finally released them as podcasts, they made it into my rotation on my commute (which had grown to 3 hours a day) and I fell in love with the comradery of the cast, the characters and the world.
When the Pandemic hit and I started working from home, we also had to stop our gaming group, so instead, wife and I started watching the old episodes of Crit Role for the first time on YouTube, since we suddenly had the extra time and not a lot of content options. It was a great escape, and wonderful to watch the show grow and the players and characters bond. And I was amazed that even though I'd listened to the campaigns before, watching them was a new and rewarding experience. Once again, it led me back to playing, when they released a free Wildemount module on Roll20, and I decided to try online play with my gaming group. Well, what was supposed to be a 3 session trial has turned into a six month campaign that goes almost every week online.
So I guess the answer is, I watch because the stories, cast and characters are entertaining (to me, way more so than most shows on network TV), but also because they inspire me to play and become a better DM. There are a ton of different styles and ways to play D&D, and watching and listening to live plays are a great way to expand your understanding of the game beyond the group you play with and make the game even more rewarding.